The conifold mirror conjecture for Calabi–Yau complete intersections in Grassmannians

Let $X_0$ be a conifold degeneration of a generic Calabi–Yau complete intersection $X\subset G(k,n)$, let $p$ be the number of nodes of $X_0$, and set $\alpha=(k-1)(n-k-1)$. Let $Y_0^*$ be the general member of the special one-parameter mirror subfamily after an MPCP-desingularization of the ambient toric variety ${\bf P}_{\nabla(k,n)}$, and let $X^*$ be a small resolution of $Y_0^*$. Conifold mirror conjecture. The variety $Y_0^*$ has the same number $p$ of nodes as $X_0$, these nodes satisfy $\alpha-p$ relations, and $X^*$ is a mirror of $X$. This predicts that the conifold transition and its mirror exchange the corresponding Hodge-theoretic changes: the toric mirror construction produces a singular member with matching nodes and relations, whose small resolution mirrors the original Grassmannian complete intersection. The source gives no resolution.