Local coherence of smooth representations for general linear groups

Determine whether the category of smooth representations $Mod(GL_n(mathfrak{F}))$ over a field of characteristic p is locally coherent for a general finite extension $\mathfrak{F}/\mathbb{Q}_p$, as suggested by the cited conjecture.

Background

The paper studies graded centers of derived categories of smooth representations and treats the locally coherent case separately. Local coherence means, in particular, that the subcategory of finitely presented smooth representations is abelian, which is needed to identify the relevant bounded derived category and compute its degree-zero graded center. The paper notes that local coherence is known for several classes of groups, including compact groups and certain special linear groups, but reports an expectation that the analogous property holds for general linear groups. Establishing this property would extend the locally coherent framework and the associated graded-center results to representations of general linear groups over non-archimedean local fields.