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The phrase "their interest in" is correct and usable in written English.
You can use it when discussing something that someone is interested in. For example, "The students demonstrated their interest in the subject by asking thoughtful questions."
Exact(59)
Their interest in community colleges impresses me.
It galvanized their interest in the objects".
But their interest in him intensified recently.
And their interest in sports has lately expanded.
Their interest in political activity was not new.
It's also a test of their interest in the position.
Winter has further eroded their interest in cave-hunting.
Democrats have certainly registered their interest in this election.
This often increased their interest in the property.
Yet their interest in it is largely about costs.
The reunion seemed to fuel their interest in her condition.
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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