Strict positivity of one-form weights for conical resolutions

Let $\pi:X\to Y$ be a conical resolution over a field of any characteristic, with

$$\mathcal{O}_Y\simeq \pi_*\mathcal{O}_X$$

and

$$R^1\pi_*\mathcal{O}_X=R^2\pi_*\mathcal{O}_X=0.$$

The multiplicative group is denoted by $\mathbb{G}_m$, and it acts on $H^0(X,\Omega_X^1)$ through the conical structure. Strict-positivity conjecture. All $\mathbb{G}_m$-weights of $H^0(X,\Omega_X^1)$ are strictly positive. In characteristic zero, the paper proves the corresponding statement under these hypotheses; it states that no counterexamples are known in characteristic $p$, which motivates the conjecture for arbitrary characteristic.