The converse to the projective support characterization for quasi-reductive supergroups

Let $G$ be a quasi-reductive supergroup, and let $M\in\operatorname{Rep}(G)$. Write $\mathfrak{g}_{\mathrm{neat}}$ for the neat locus in the odd part of the Lie superalgebra, and $\operatorname{supp}(M)$ for the support of $M$. Support-projectivity conjecture. If

$$\operatorname{supp}(M)=\mathfrak{g}_{\mathrm{neat}},$$

then $M$ is projective. This conjectures the converse of the preceding projective-support characterization; it is posed for quasi-reductive supergroups and is presented as an open question in the source.