SEOUL, SOUTH KOREA (NYTIMES) - If there is anyone keeping his fingers crossed for President Donald Trump and the North Korean leader Kim Jong Un to agree on how to denuclearise the North when they meet in Vietnam this month, it is President Moon Jae-in of South Korea.

With no quick fix available for South Korea's stubborn economic troubles, Mr Moon's best chance for reversing his falling approval ratings rests on whether he can jump-start his signature policy of helping advance the North's denuclearisation and improving inter-Korean ties.

But that depends on Mr Trump and Mr Kim striking a denuclearisation deal when they meet on Feb 27-28 in Hanoi, the Vietnamese capital, that is significant enough for Washington and the United Nations to ease sanctions and create room for Mr Moon to push his ambitious plans for economic cooperation with North Korea.

"But if the second US-North Korea summit fails, there will be more criticism blaming President Moon," said Dr Koh Yu-hwan, a professor of North Korean studies at Dongguk University in Seoul.

"There will be more critics accusing him of having been too optimistic and naïve, misreading North Korea's intentions and misrepresenting them to the Americans."

Since his election in 2017, Mr Moon has dedicated himself to mediating between Washington and Pyongyang, tirelessly selling Mr Trump on the merits of negotiating with Mr Kim.

In doing so, he has bet so heavily on Mr Trump and Mr Kim reaching a denuclearisation deal that his own political fortunes at home have become increasingly tied to the whims of the two unpredictable leaders.

Mr Moon's approval ratings soared higher than 80 per cent last spring when he met Mr Kim twice on the inter-Korean border to help defuse a possible military confrontation between the United States and North Korea.

That set the stage for the historic June summit meeting between Mr Kim and Mr Trump in Singapore.

But Mr Moon's ratings have since plummeted below 50 per cent amid stalled talks between Washington and Pyongyang over how to carry out the vague promise Mr Trump and Mr Kim made in Singapore to "work toward complete denuclearisation of the Korean Peninsula" and establish a permanent peace between their countries.

Mr Moon flew to Pyongyang in September to meet Mr Kim again to break the logjam between Washington and Pyongyang.

But the deadlock remains, prompting Mr Moon's critics to question whether he has oversold Mr Kim's willingness to denuclearise.

When he met Mr Moon in September, Mr Kim promised to become the first North Korean leader to visit Seoul.

But with his negotiations with Washington stalled, the year ended without Mr Kim visiting as Mr Moon said he would, further raising doubts about Mr Moon's influence over Mr Kim.

While Mr Moon remained preoccupied with North Korea, domestic affairs turned against him.

The popularity of his governing Democratic Party has plunged as its politicians have been tainted by #MeToo accusations and other scandals.

South Koreans in their 20s, a traditional support base for progressives like Mr Moon, have been fast losing faith in his government because of his inability to create more jobs, according to the polling company Realmeter.

Older, more conservative South Koreans have rallied in central Seoul almost every weekend in recent months, criticising as dangerous Mr Moon's economic policies and his rapprochement efforts with the North.

But Mr Moon remains convinced that Mr Trump's strong desire to achieve something none of his predecessors could accomplish - ending the North Korean nuclear crisis - and Mr Kim's desperate need to improve his country's economy can create a singular opportunity for both Koreas.

Mr Trump and Mr Kim, he says, can denuclearise the Korean Peninsula and end the tensions there, opening the way for economically integrating the impoverished North, with its cheap labour, and the South, whose slowing economy needs a new source of growth.

"We have never had an opportunity like this since the Korean War ended in an armistice in 1953, and we should not miss it because it will never come again," Mr Moon said last month.

Long-time North Korea observers saw the current stalemate coming when Mr Trump and Mr Kim ended their Singapore meeting without sorting out key details, especially what is meant by the "complete denuclearisation of the Korean Peninsula".

In the past, the North used that term when it argued that it would give up its nuclear weapons only when the United States ceased hostilities, including ending its military presence in South Korea.

Mr Moon says Mr Kim has made a strategic decision to give up his nuclear weapons in a "verifiable and irreversible way" and focus on rebuilding his country's economy should Washington take corresponding actions, like easing sanctions, to prove that it is no longer hostile.

He also said last month that there was "no difference" between the United States and North Korean definitions of "complete denuclearisation", saying it meant dismantling all North Korean nuclear weapons and fissile materials and their production facilities.

"First, we ought to take him at his word," General Vincent Brooks, who retired as commander of the US military in South Korea in November, told PBS NewsHour last month, referring to Mr Kim.

"And that's not an easy thing to accept, especially given the track record of North Korea. But this is a new leader in North Korea."

Mr Trump, who in 2017 threatened to unleash "fire and fury" against North Korea and derided Mr Kim as "Little Rocket Man" because of his provocative missile tests, now promises to help rebuild the country's economy if it denuclearises.

"North Korea will become a different kind of rocket - an economic one!" he said on Twitter this month.

But last month, Mr Trump's director of national intelligence, Mr Dan Coats, told Congress that North Korea was "unlikely to completely give up its nuclear weapons and production capability because its leaders ultimately view nuclear weapons as critical to regime survival".

Nevertheless, Washington's special representative for North Korea, Mr Stephen Biegun, spent three days in Pyongyang this month exchanging "detailed and exhaustive" lists of what the two governments wanted from each other to make denuclearisation possible, said Mr Kim Eui-kyeom, Mr Moon's spokesman, after Mr Biegun briefed South Korean officials on his trip.

But with the Hanoi summit meeting less than a week away, South Korean officials said, the two sides are still bargaining to settle the hardest part of their negotiations: figuring out what specific actions they could take in order to start denuclearisation.

All this makes the Hanoi meeting as much a gamble for Mr Moon as for Mr Trump and Mr Kim.

Unlike Mr Kim, who has hedged his bets by improving ties with China, Mr Moon has put all his eggs in Mr Trump's basket, analysts said.

"Presidents Moon and Trump are in the same boat on this," said Mr Kim Sung-han, a former vice foreign minister of South Korea who teaches at Korea University in Seoul.