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The phrase "a problem that remains" is correct and usable in written English
It can be used to refer to an issue or challenge that continues to exist despite efforts to resolve it. Example: "Despite the progress we have made, there is still a problem that remains in our approach to sustainability."
Exact(22)
A problem that remains uncertain or ambiguous will remain as such until it is defined, regardless of the approach that is used to address it.
A problem that remains is the Y.M.C.A.'s name.
A problem that remains unresolved is background noise.
"This is a problem that remains," he said.
However, whether the modification has any influence on the drug itself is still a problem that remains to be solved.
And that has meant embracing plastic in a big way - a problem that remains despite manufacturers moving to reduce packaging in 2009.
Similar(38)
To ensure a book's long life, authors should pick a big problem that remains unsolved and big.
Heavily traveled, these roads are subject to traffic gridlock, a pervasive problem that remains unresolved.
Many people are concerned that they have a thyroid problem that remains undiagnosed.
Attribution of the tidal spectrum at 110 km to specific sources and propagation conditions at lower altitudes is a difficult problem that remains to be solved.
It's a toxic problem that remains largely submerged.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com