The weak Landau–Ginzburg model formula for cyclic covers

Let $Y$ be a smooth Fano variety of index $r$ and $D_Y\thinspace\subseteq\thinspace Y$ a smooth anti-canonical divisor. Let $X$ be the $r$-fold cyclic covering of $Y$ branched along $D_Y$, and let $f_Y$ be a weak Landau–Ginzburg model for $Y$. Weak Landau–Ginzburg model formula. The Laurent polynomial

$$f_Y^r-c_0(f_Y^r)$$

is a weak Landau–Ginzburg model for $X$. This is presented as an algebro-geometric consequence of the open Gromov–Witten formula and relates cyclic covers to mirror constructions; the source does not state whether the claim has been resolved.