Let $X$ be a smooth projective variety defined over a field $k$. In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:

$(\ast) \quad$ $\mathrm H^0 \big( X, \Omega_X^p \otimes \mathscr L^{-1} \big) = 0$ for all $p < \dim X$ and ample line bundles $\mathscr L$.

I am interested in what happens in positive characteristic.

**Question 1:** Is there a counterexample to $(\ast)$ when $\mathrm{char}(k) > 0$?

Since the statement is obviously related to Kodaira vanishing, I would guess that a counterexample is already known. The following question, however, might be harder:

**Question 2:** Is there a counterexample as above, but additionally with $K_X \sim_{\mathbb Q} 0$ or $X$ Fano?

Actually, a log version would also be sufficient, with $\Omega_X^p$ replaced by $\Omega_X^p(\log D)$, where $(X, D)$ is an snc pair (and in the second question, $K_X + D \sim_{\mathbb Q} 0$ or $(X, D)$ log Fano).