Strong liftability conjecture for cyclic covers over smooth projective toric varieties

Let $X$ be a smooth projective toric variety, and let $\mathcal L$ be an invertible sheaf on $X$. Let $N$ be a positive integer prime to $p$, and let $D$ be an effective divisor on $X$ such that

$$\mathcal L^N=\mathcal O_X(D)$$

and $\operatorname{Sing}(D_{\rm red})=\emptyset$. Let $\pi:Y\rightarrow X$ be the cyclic cover obtained by taking the $N$-th root out of $D$.

Strong liftability conjecture. The scheme $Y$ is a smooth projective scheme which is strongly liftable over $W_2(k)$.

This conjecture proposes that the strong liftability result known for cyclic covers of projective spaces extends to cyclic covers over all smooth projective toric varieties. Ordinary liftability in this setting has already been proved, but strong liftability remains open in the supplied source.