The toric compactification conjecture for mirrors of Calabi–Yau complete intersections

Let $V$ be a Calabi–Yau complete intersection defined by hypersurfaces $V_1,\ldots,V_r$, and let ${\bf T}$ be the associated $(d+r)$-dimensional algebraic torus. For the subsets $E_1,\ldots,E_r$ in the toric construction, let $P_{E_i}(X)$ denote the corresponding affine hypersurface equations.

Toric compactification conjecture. The mirror Calabi–Yau varieties with respect to $V$ are Calabi–Yau compactifications of the complete intersection in ${\bf T}$ defined by

$$P_{E_1}(X)=\cdots=P_{E_r}(X)=0.$$

This is the proposed geometric realization of the mirror as a compactification of an affine torus complete intersection; the source gives no general resolution.