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The phrase 'remained unsolved' is a correct and usable part of any sentence in written English.
You can use it when referring to a problem, dilemma, or mystery that has not been resolved. For example, "The mystery of what happened to Amelia Earhart remained unsolved."
Exact(60)
The Eastburn murders, meanwhile, remained unsolved.
But in public, the case remained unsolved.
In her case the mystery remained unsolved.
The case has remained unsolved ever since.
The murder has remained unsolved for 67 years.
Juliane Willuhn, the center's director, said the crime remained unsolved.
But the mysterious nature of the attack remained unsolved.
And there things have remained, unsolved – until now.
In the end, the issue remained unsolved, the legislation dead.
The chronic foreign policy problems remained unsolved and unchanged.
Yet the mystery about the heterophilic bond between these two proteins remained unsolved.
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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