Finite generation of subobjects in the geometric Hecke category

Let $\Catgeom$ be the abelian category of objects equipped with compatible Hecke isomorphisms, and call an object finitely generated if it admits a surjection from an object of the form $\CS^1\star \Rg$ with $\CS^1\in \Pervgr$. Finite-generation conjecture. A sub-object of a finitely generated object of $\Catgeom$ is finitely generated. This is a noetherianity-type assertion for the geometric Hecke category; the supplied text gives no resolution or supporting evidence.