After receiving another letter from Kim Jong Un, President Donald Trump on Thursday thanked the North Korean dictator for his "kind action" in handing over the possible remains of U.S. soldiers from the Korean War.

But even as Trump praised Kim "for keeping his word," there are deep doubts about the strength of the document the two men signed in Singapore and North Korea's understanding of the "complete denuclearization" the U.S. is demanding and North Korea agreed to.

The White House received the letter on August 1, according to press secretary Sarah Sanders, who said the president was responding in his own letter to be delivered shortly. The two leaders have exchanged a series of letters now, usually hand-delivered by their top negotiators -- Secretary of State Mike Pompeo and Kim Yong Chol, Vice Chairman of North Korea's ruling Workers' Party Central Committee.

Sanders said the letters "addressed their commitment from their joint statement that was made at the Singapore summit, and they're going to continue working together towards complete and total denuclearization."

Thank you to Chairman Kim Jong Un for keeping your word & starting the process of sending home the remains of our great and beloved missing fallen! I am not at all surprised that you took this kind action. Also, thank you for your nice letter - l look forward to seeing you soon!

— Donald J. Trump (@realDonaldTrump) August 2, 2018

Trump added that he looked "forward to seeing you soon!" although it's unclear what he was referring to. Sanders told reporters Thursday that no meeting is "locked in or finalized," but they are open to one. There has also been speculation that Trump might allow Kim to travel to New York for the United Nations General Assembly in September.

Trump's praise for Kim centered on returning remains of American soldiers killed during or missing from the Korean War -- one of the four provisions in the joint declaration the two signed in Singapore in June. Fifty-five containers carrying what are believed to be American remains arrived in Hawaii Wednesday. Vice President Mike Pence oversaw the solemn ceremony as the caskets were carried off U.S. military aircraft and onto U.S. soil at Joint Base Pearl Harbor-Hickam.

"I know that President Trump is grateful that Chairman Kim has kept his word, and we see today as tangible progress in our efforts to achieve peace on the Korean Peninsula," Pence said, but, "Our work will not be complete until all our fallen heroes are accounted for and home."

The 55 remains are just a fraction of the Americans missing from the Korean War. The Pentagon's POW/MIA Accounting Agency estimates there are 7,697 soldiers unaccounted for, with approximately 5,300 expected to be located in North Korea. It will also take years for forensic teams to now try to identify the 55 remains and return them to the closest next of kin.

While seen as an important first step by North Korea to fulfill its obligations under Trump and Kim's joint declaration, the regime has not yet taken tangible steps towards the U.S.'s broader goal, of dismantling the country's nuclear weapons program. Instead, there have been signs that North Korea is moving ahead with key aspects of it.

Satellite images published Monday appear to confirm the Washington Post's report that North Korea is continuing to construct one or two intercontinental ballistic missiles. Last week before the Senate Foreign Relations Committee, Pompeo confirmed that North Korea continues to produce fissile material -- the fuel for nuclear bombs. But he insisted that there had been progress since the summit and the administration has not been "taken for a ride."

Sanders wouldn't say Thursday whether the president was satisfied with the progress so far, only that he wouldn't be satisfied until North Korea fully denuclearizes.

The U.S. is also concerned about North Korea evading sanctions, especially with the assistance of China and Russia. A senior State Department official said Thursday, "We do have concerns about North Korea bypassing some of those sanctions, not adhering to its own obligations." Two weeks ago, Pompeo joined U.S. Ambassador to the U.N. Nikki Haley to urge the two countries to stop smuggling oil to the regime.

Pompeo is in Southeast Asia ahead of a major summit in Singapore, where it's possible he may meet with the North Korean Foreign Minister, according to a senior State Department official. The last first in-person meeting between the two sides was on July 27, according to a State Department official, who would not say whether that addressed denuclearization or the repatriation of remains.

But it would also be Pompeo's first meeting with North Korean officials since he visited Pyongyang at the beginning of July. Those two days of meetings ended without a one-on-one between Pompeo and Kim Jong Un, who instead was inspecting a potato farm, and with a blistering statement from North Korea that criticized the U.S.'s "robber-like" demands.

Still, since then, the two sides have had "constructive" conversations, according to State Department spokesperson Heather Nauert, which continue. The administration points to the destruction of some key facilities at the missile test site as a sign of success, saying Kim personally promised that to Trump in Singapore. But experts say the site could be easily reassembled and is no longer necessary for North Korea's nuclear weapons program.

While the development of new missiles or fissile material production has been conveyed as North Korea "cheating," it's unclear if that's what North Korea believes it agreed to at the Singapore summit. The State Department declined to say whether ICBM development would violate the declaration on Tuesday, but Pompeo told the Senate last week that North Korea agrees to the U.S.'s definition of "complete denuclearization," which includes their weapons systems.

But Jenny Town, an analyst with the North Korea watchers at 38 North, told ABC News Tuesday that North Korea never agreed to an immediate halt of all activity or to specifically shut down this site.

"The work that's being done is not work that's suddenly started," she added, noting that if North Korea ramped up a program or start something new, that would be cause for concern.

There are still no international inspectors on the ground in North Korea to monitor what is happening.

ABC News's Ben Gittleson contributed to this report.