Hilbert's Sextic Conjecture for the Valentiner action

Let $T \rightarrow \operatorname{Spec}(\mathbb C(x,y))$ be the $A_6$-torsor associated to the Valentiner action $A_6 \curvearrowright \mathbb P^2_{\mathbb C}$. For a smooth, irreducible, generically free $A_6$-curve $X$, write $\prescript{T}{}{X}$ for the twist of $X$ by $T$, and let $\mathbb C(x,y)^{(1)}$ denote the specified extension appearing in the conjecture.

Hilbert's Sextic Conjecture. For any smooth, irreducible, generically free $A_6$-curve $X$,

$$\prescript{T}{}{X}\left(\mathbb C(x,y)^{(1)}\right)=\emptyset.$$

This is a reformulation of Hilbert's sextic conjecture in terms of the Valentiner $A_6$-action and generalized versality. The supplied text gives no resolution status, so the conjecture is recorded as open.