Gepner-type stability conjecture for smooth quintic threefolds

Let $X\subset\mathbb{P}^4$ be a smooth quintic 3-fold, and assume the strong Bogomolov–Gieseker conjecture used to construct the heart $\mathcal{A}_G$ and central charge $Z_G^{\dag}$. Let $\mathrm{ST}_{\mathcal{O}_X}$ denote the spherical twist by $\mathcal{O}_X$, and let $\otimes\mathcal{O}_X(1)$ denote tensoring by the hyperplane line bundle. Gepner-type stability conjecture. The pair

$$(Z_G^{\dag},\mathcal{A}_G)$$

determines a Gepner-type stability condition on $D^b\operatorname{Coh}(X)$ with respect to $(\mathrm{ST}_{\mathcal{O}_X}\circ\otimes\mathcal{O}_X(1),2/5)$. This would provide a stability condition compatible with the Gepner symmetry for the quintic threefold; its validity depends on the strong Bogomolov–Gieseker conjecture assumed in the statement.