Hom-finiteness conjecture for homotopy categories of supergroup representations

Let $G$ be an algebraic supergroup with reductive even part, let $\mathcal T=\operatorname{Rep}_k(G)$, and let $\mathcal C=\operatorname{Rep}(G)^\infty$. For objects $X,Y\in\mathcal T$, write $[X,Y]$ for the morphism space in the associated homotopy category. Assume conditions A-1–A-4 hold. Hom-finiteness conjecture. For all objects $X,Y\in\mathcal T$,

$$\dim_k([X,Y])<\infty.$$

This conjecture asserts finite-dimensionality of all morphism spaces in the homotopy category; the source also notes that it is enough to establish finite-dimensionality for morphisms from irreducible objects to the tensor unit. A special case for $(P(m|n)^+,GL(m|n))$ is stated as a lemma.