Optimistic conjecture for quotient EPW sextics

Let $\mathcal C$ be the class of varieties whose Chow rings admit the multiplicative bigrading described in Beauville's weak splitting problem. Let $X$ be an EPW sextic, meaning a special sextic $X\subset\mathbb{P}^5(\mathbb{C})$, and assume that $X$ is a quotient variety

$$X=X_0/G$$

with $X_0$ smooth and $G\subset\operatorname{Aut}(X_0)$ a finite group. Optimistic conjecture. Then $X\in\mathcal C$. This is motivated by the expected weak splitting behavior of Calabi–Yau and hyperkähler varieties and is open.