Cycle-class injectivity for powers of quotient EPW sextics

For a projective quotient variety $X$, let $E^\ast(X^r)\subset A^\ast(X^r)$ be the $\mathbb{Q}$-subalgebra generated by pullbacks of elements of $A^1(X)$ and $A^2(X)$, pullbacks of the diagonal $\Delta_X$, and pullbacks of the small diagonal $\Delta^X_{sm}$. EPW cycle-class conjecture. Let $X$ be an EPW sextic as in the optimistic conjecture, and let $r\ge 1$. The restriction of the cycle class map

$$E^i(X^r)\to H^{2i}(X^r,\mathbb{Q})$$

is injective for all $i$. This is a concrete, falsifiable form of the expected weak splitting property for quotient EPW sextics and remains open.