Non-abelian Galois-cover generalization of the rational-cover rigidity theorem

Let $V$ be a divisorially canonical Fano variety satisfying the hypotheses of Theorem 1, and let $X\dashrightarrow V$ be a rationally connected Galois rational cover. A Galois group here means the Galois group of the corresponding extension of function fields $\mathbb C(V)\subset\mathbb C(X)$. Non-abelian Galois-cover conjecture. The conclusion of Theorem 1 should remain valid without assuming that the Galois group is abelian; equivalently, there are no nontrivial rationally connected Galois rational covers of $V$ with an arbitrary finite Galois group. The paper explains that the proposed extension is motivated by the invariance of the ramification divisor under the Galois-group action, but it is not proved there.