Fukaya–Yamaguchi conjecture on bounded virtual abelianness for finite fundamental groups

Let $(X,d,\mathfrak{m})$ be an $RCD^{\ast}(K,N;D)$ space. If its fundamental group $\pi_1(X)$ is finite, then Fukaya–Yamaguchi conjecture. It contains an abelian subgroup $A$ of index

$$[\pi_1(X):A]\leq C(N,KD^2).$$

This is another formulation of the problem of obtaining uniform virtually abelian control without assuming non-collapsing; the source leaves it as an open problem.