Refined Weil–Petersson conjecture for the stringy Kähler moduli space

Let $X$ be a projective Calabi–Yau $n$-fold, let $\mathcal{M}_{\mathrm{Kah}}(X)$ be its stringy Kähler moduli space, and let $\mathcal{D}_X=\mathrm{D^bCoh}(X)$. Let $\mathrm{Stab}_{\mathcal{N}}^+(\mathcal{D}_X)$ denote the relevant component of the numerical stability-condition space, and let $K_{\mathrm{WP}}$ be the Weil–Petersson potential. For a mirror manifold $Y$, write $\mathcal{M}_{\mathrm{cpx}}(Y)$ for its complex moduli space. Refined Weil–Petersson conjecture. There exists an embedding

$$\iota:\mathcal{M}_{\mathrm{Kah}}(X)\hookrightarrow\mathrm{Aut}(\mathcal{D}_X)\backslash\mathrm{Stab}_{\mathcal{N}}^+(\mathcal{D}_X)/\mathbb{C}.$$

The complex Hessian of $\iota^*K_{\mathrm{WP}}$ defines a non-degenerate Kähler metric on $\mathcal{M}_{\mathrm{Kah}}(X)$, and this metric is identified under the mirror map $\mathcal{M}_{\mathrm{Kah}}(X)\cong\mathcal{M}_{\mathrm{cpx}}(Y)$ with the Weil–Petersson metric on $\mathcal{M}_{\mathrm{cpx}}(Y)$. When $n=3$, the image is locally a holomorphic Legendre variety. The authors state that the conjecture holds for elliptic curves and mention supporting evidence, but the general claim remains open; defining the stringy Kähler moduli space is identified as a major difficulty.