The thickness conjecture for unstable modules with finite injective hulls

Let $\mathcal U$ be the category of unstable modules, and consider the full subcategory whose objects are unstable modules whose injective hull is a finite direct sum of indecomposable injective unstable modules. Thickness conjecture. This full subcategory of $\mathcal U$ is thick. In particular, quotients of such objects have the same property. The claim is motivated by Steven Sam's proof of the artinian conjecture and related work, but the source presents it as a conjecture and does not state a resolution.