Gerbe-independence conjecture for virtual fundamental classes

Let $W$ be the coarse projective variety in the KSBA moduli problem, let ${\mathfrak W}\to W$ be a $\mu_N$-gerbe, and let $M$ be the associated KSBA moduli space. Suppose that the virtual fundamental class is defined by pushing forward the virtual class from the moduli space of index one covers:

$$[M,{\mathfrak W}]^{\operatorname{vir}}:=f_*[M^{\operatorname{ind}}]^{\operatorname{vir}}.$$

Here $N$ is sufficiently large and divisible by the orders of the cyclic groups occurring in the index one covering DM stacks. Gerbe-independence conjecture. For large divisible $N\gg0$, the class $[M,{\mathfrak W}]^{\operatorname{vir}}$ is independent of the choice of the $\mu_N$-gerbe ${\mathfrak W}\to W$.

The conjecture asks whether the enumerative virtual class descends independently of the auxiliary gerbe used to construct it. The supplied text explicitly says that this independence is currently unknown.