The non-increasing order cover conjecture for torsion classes

Let $A$ be the algebra under consideration, let $g\text{-}\mathrm{TF}(A)$ denote the $g$-vector fan, let $\mathsf{binuc}(A)$ denote the set of binuclear torsion classes, and let $\leq_{\mathrm{NI}}$ be the non-increasing order on torsion classes. For $[\theta]\in g\text{-}\mathrm{TF}(A)$, write $\mathsf{\int}[\theta]$ for the interior of its corresponding face.

Non-increasing cover conjecture. If

$$[\theta]\in g\text{-}\mathrm{TF}(A),\qquad I\in\mathsf{binuc}(A),\qquad \mathsf{\int}[\theta]\leq_{\mathrm{NI}} I,$$

then there exists $[\rho]\in g\text{-}\mathrm{TF}(A)$ such that

$$[\theta]<\!\!\!\!\cdot_{\mathrm{NI}}[\rho]\leq_{\mathrm{NI}} I.$$

This would extend the preceding cover theorem from the facial semistability order to the non-increasing order, providing an intermediate $g$-vector face between $[\theta]$ and any suitable binuclear torsion class $I$.