Koszulity conjecture for the locally finite endomorphism algebra of a projective generator

Let $\mathcal B$ be a chosen block, let $P=\bigoplus_{\lambda\in\Lambda(\mathcal B)}P(\lambda)$ be a minimal projective generator of $\mathcal B$, and let

$$\operatorname{End}^{\operatorname{fin}}_{\mathcal F}(P)=\bigoplus_{\lambda\in\Lambda(\mathcal B)}\operatorname{Hom}_{\mathcal F}(P(\lambda),P)$$

denote its locally finite endomorphism algebra. Koszulity conjecture. The algebra $\operatorname{End}^{\operatorname{fin}}_{\mathcal F}(P)$ is Koszul. This would establish that the grading arising from the diagrammatic algebra is a Koszul grading, extending the known analogy with the general linear supergroup case; the context states that the expectation is readily verified for $\operatorname{OSp}(3|2)$, but gives no general resolution.