Mirror-family conjecture for nef-cone faces

Let $(X,C)$ be a lattice-polarized weak del Pezzo pair of degree $d$, let $\mathcal Y$ be its mirror family, and let $\alpha_i$ be the simple root associated to a face $F_i$ of $\operatorname{Nef}(X)$ with $[C]\subset F_i$. Let $\mathcal A_d(x_i)$ denote the variables corresponding to the marking-equivalence class of this root.

Mirror-family conjecture. For each such face $F_i$, there exists a codimension-one subfamily $\mathcal Y_i$ of $\mathcal Y$ whose members are $\langle\alpha_i\rangle$-polarized rational elliptic surfaces of type $d$, given by an equation $f_i(\mathcal A_d(x_i))=0$. Moreover, if $S=\bigcap_{i\in I}F_i$ is a subface, then

$$\mathcal Y_S:=\bigcap_{i\in I}\mathcal Y_i$$

is a family of $\langle\alpha_i:i\in I\rangle$-polarized rational elliptic surfaces of type $d$.

This formalizes how faces of the nef cone should correspond to lattice-polarized subfamilies in the mirror family. The source presents the construction as conjectural and gives no resolution.