Finiteness of the topological fundamental group of an irreducible symplectic regular locus

Let $X$ be an irreducible symplectic variety, and let $X^{\mathrm{reg}}$ denote its regular locus. Write $\pi_1(X^{\mathrm{reg}})$ for the topological fundamental group of $X^{\mathrm{reg}}$.

Finiteness conjecture. The group $\pi_1(X^{\mathrm{reg}})$ is finite:

$$|\pi_1(X^{\mathrm{reg}})|<\infty.$$

The algebraic fundamental group of the regular locus is known to be finite, but the corresponding assertion for the topological fundamental group is posed as an open question.