The flip-folded transposition-mirror conjecture for non-Fano toric hypersurfaces

Let $Z_f\subset X$ be a Calabi–Yau hypersurface in a non-Fano toric variety $X$ whose spanning polytope $\mathcal D^\star(X)$ is non-convex. Let $\widetilde X$ be the toric space associated with the flip-folded spanning multitope \mathcal D^\star(\widetilde X)=(\mathcal D^\star(\widetilde X))^\widetilde{\ }\subset\mathcal D(X), and let $Z_{f^{\mathrm T}}\subset\widetilde X$ be the transposed hypersurface.

Flip-folded transposition-mirror conjecture. (1) The transposition mirror may be found as the transposed hypersurface $Z_{f^{\mathrm T}}$ in $\widetilde X$. (2) The toric space $\widetilde X$, and possibly the mirror hypersurface $Z_{f^{\mathrm T}}$, are pre-complex: their complex structures degenerate at isolated locations corresponding to the flip-folded elements of $\mathcal D^\star(\widetilde X)$.

This conjecture extends the transposition-mirror construction to non-convex spanning polytopes and predicts isolated complex-structure obstructions. The source supplies no proof, so it remains open.