The converse vanishing theorem in Verdier quotients of stable categories

Assume $(A,\mathfrak m)$ is a complete local complete intersection of dimension $d$ and codimension $c$, with algebraically closed residue field $k=A/\mathfrak m$. Let $X$ be an algebraic set in $\mathbb P^{c-1}(k)$, let $\mathcal T_X=\operatorname{\underline{CM}}(A)/\mathcal S_X$, and let $M,N$ be maximal Cohen–Macaulay $A$-modules. Write $\mathcal V_X(M)=\mathcal V^*(M)\setminus X$. The converse vanishing conjecture. If

$$\mathcal V_X(M)\cap\mathcal V_X(N)=\emptyset,$$

then

$$\operatorname{Hom}_{\mathcal T_X}(M,\Omega^{-n}(N))=0\quad\text{for all }n\gg0.$$

The paper proves this implication when $M$ or $N$ is essentially disjoint from $X$, while the conjecture asks for it without that additional assumption. The forward implication is established generally, so the open issue is precisely the converse.