Lift conjecture for universal torsors of projective Calabi–Yau manifolds

Let $X$ be a projective Calabi–Yau manifold, meaning that its canonical bundle is torsion and

$$H^1(X,\operatorname{\mathcal O}_X)=0,$$

and let $\mathcal U$ be a universal torsor for $X$. Lift conjecture. The action of $\operatorname{Aut}(X)$ on $X$ should lift to an action on $\mathcal U$ after possibly passing to a finite-index subgroup. More generally, the same should hold for the group of birational automorphisms of $X$. The conjecture concerns equivariant structures on universal torsors in the Calabi–Yau setting; the source presents it as an expected statement, without giving a resolution.