Geometric Jordan conjecture for the Cremona group

Let $n$ be a positive integer, let $X$ be an $n$-dimensional rational variety, and let $G$ be a finite group acting on $X$. Geometric Jordan conjecture. There exists a constant $c(n)$ depending only on $n$ such that $G$ contains a normal abelian subgroup $A$ of index at most $c(n)$ and

$$A\leqslant \mathbb{G}_m^k\leqslant \operatorname{Bir}(X)$$

for some $k$. The conjecture asks whether the finite-group Jordan property can always be realized by a subgroup contained in an algebraic torus of the birational automorphism group. The conjecture remains open in dimension three and higher.