Gepner-type stability condition conjecture for chain-type invertible polynomials

Let $f_n$ be the chain-type invertible polynomial, let $S_n$ and $L_{f_n}$ be the associated ring and grading group, and let $\omega_1,\dots,\omega_n$ be positive rational numbers satisfying

$$f_n({\mathbf e}[\omega_1]z_1,\dots,{\mathbf e}[\omega_n]z_n)=f_n(z_1,\dots,z_n).$$

Write $\widetilde f_n$ for the mirror polynomial, $E_j$ for the relevant objects, $d_n$ for the degree parameter, and $d{\bf x}=dx_1\wedge\cdots\wedge dx_n$. The Gepner-type stability condition conjecture. There exists a Gepner-type stability condition $\sigma$ on ${\rm HMF}^{L_{f_n}}_{S_n}(f_n)$ with respect to the auto-equivalence $(\vec z_1)$ and ${\bf e}[1/d_n]\in\mathbb C$ such that

$$(\vec z_1).\sigma=\sigma.{\bf e}\left[\frac{1}{d_n}\right],$$

and its stability function $Z_\sigma:K_0({\rm HMF}^{L_{f_n}}_{S_n}(f_n))\longrightarrow\mathbb C$ is given by

$$Z_\sigma(E_j):=\begin{cases}\displaystyle \frac{1}{(2\pi\sqrt{-1})^n}{\mathbf e}\left[-\frac{j-1}{d_n}\right]\prod_{i=1}^{m}\left(1-{\bf e}[-\omega_{2i-1}]\right)\displaystyle\int_{(\mathbb R_{\geq0})^n}e^{-\widetilde f_n({\bf x})}d{\bf x},&n=2m-1,\\[6pt]\displaystyle \frac{1}{(2\pi\sqrt{-1})^n}{\mathbf e}\left[\frac{j-1}{d_n}\right]\prod_{i=1}^{m}\left(1-{\bf e}[-\omega_{2i}]\right)\displaystyle\int_{(\mathbb R_{\geq0})^n}e^{-\widetilde f_n({\bf x})}d{\bf x},&n=2m. \end{cases}$$

Here the stability condition is expected to arise from the oscillatory integral and homological mirror symmetry; the source does not state that this conjecture has been proved.