The mirror-family conjecture for Calabi–Yau complete intersections in projective space

Let $V$ be a $d$-dimensional Calabi–Yau complete intersection of $r$ hypersurfaces in projective space, and let $V'$ be the associated one-parameter family of $d$-dimensional Calabi–Yau compactifications constructed from the corresponding affine torus complete intersections.

Mirror-family conjecture. The one-parameter family of $d$-dimensional varieties $V'$ yields the mirror family for $V$.

This proposes an explicit construction of mirrors for Calabi–Yau complete intersections, but the source gives no resolution of the claim.