Let $P \subset \mathbb{R}^{n-1}$ be a lattice polytope of dimension $n-1$ and let $\sigma \subset \mathbb{R} \times \mathbb{R}^{n-1}$ be the cone over $1 \times P$. To the cone $\sigma$, we may associate an n-dimensional Gorenstein affine toric variety $X_{\sigma}$. Fix an integer $k\geq 0$, to such a variety we may assign a sheaf of reflexive differential $ k$ forms $$ \tilde{\Omega}_{X_{\sigma}}^k= (\Omega_{X_{\sigma}}^k)^{\ast \ast}$$ Equivalently, we can consider the pushforward $i_*(\Omega_U^k)$ of from the smooth locus $U$.
Suppose that $X_\sigma$ admits a toric resolution $\pi: X \to X_\sigma$ coming from a regular subdivision of $P$.
Question A. Is it true that $\pi_*(\Omega^k_{X})= \tilde{\Omega}_{X_{\sigma}}^k ?$
Question B. Do we have $R^p\pi_*(\Omega^k_{X})=0$ for $p>0$?
Remarks. This is stated in Cox-Little-Schenck's book for $k=n$. If I am reading the main theorem of https://arxiv.org/pdf/1003.2913.pdf correctly, the answer to Question A. is yes (apparently toric singularities are "klt"). However, this paper is rather advanced and the situation I'm interested for now is very explicit.
