Generic semistability of projected push-forwards for cubic and Gushel–Mukai fourfolds

Let $X$ be a cubic fourfold or a GM fourfold, and let $j\colon Y\hookrightarrow X$ be its smooth hyperplane section. Let $\sigma_Y$ be a Serre-invariant stability condition on $\mathcal{K}u(Y)$, and let $M^Y_{\sigma_Y}(a,b)$ denote the moduli space of S-equivalence classes of $\sigma_Y$-semistable objects of class $a\lambda_1+b\lambda_2$. If $\gcd(a,b)=1$, then there exist a stability condition $\sigma_X\in\operatorname{Stab}^{\circ}(\mathcal{K}u(X))$ generic with respect to $a\Lambda_1+b\Lambda_2$ and an open smooth subscheme $U\subset M^Y_{\sigma_Y}(a,b)$.

Generic projected-push-forward conjecture. For every $[E]\in U$, the object $\operatorname{pr}_X(j_*E)$ is $\sigma_X$-semistable.

This proposes a generic stability result for projected push-forwards from smooth hyperplane sections of cubic and GM fourfolds. The source supplies no evidence of a resolution.