Toda's Gepner-point existence conjecture for graded matrix factorizations

Let $f$ be the homogeneous polynomial defining the graded matrix-factorization category $\operatorname{HMF}(f)$, and let $h$ be its degree. Let $Z_G$ be the central charge symbolically defined by

$$Z_G(P^\bullet)=\operatorname{sTr}(e^{2\mathbf{i}\pi/h}:P^\bullet\to P^\bullet).$$

A stability condition on this category is a pair $(Z,\mathcal{P})$, and the grade-shift autoequivalence is denoted by $\overline{\tau}$. Toda's conjecture. There is a stability condition $\sigma_G=(Z_G,\mathcal{P}_G)$ whose central charge is $Z_G$, and it satisfies the Gepner-type equation

$$\overline{\tau}(\sigma)=\frac{2}{h}\cdot\sigma.$$

Equivalently, the equation admits a solution in the $\mathbb{C}$-orbit $\mathbb{C}\cdot\sigma_G$. The conjecture predicts a distinguished Gepner point for matrix-factorization categories; the source attributes it to Toda and does not give evidence of a general resolution.