The symmetric powers conjecture for $\mathbb{D}$-singularities

Let $X$ be a smooth variety over $k$, and let $m\geq 1$. The symmetric power $\operatorname{Sym}^m X=X^m/S_m$ is the quotient by the symmetric group. A variety has $\mathbb{D}$-singularities when its class satisfies the corresponding condition defined using the involution $\mathbb{D}$ in the graded Grothendieck ring. Symmetric powers conjecture. The symmetric power $\operatorname{Sym}^m X$ has $\mathbb{D}$-singularities.

This is motivated by the theorem that the class of $\operatorname{Sym}^m X$ lies in the relevant special submodule up to torsion, and the conjecture asks whether the geometric $\mathbb{D}$-singularity property always follows. It remains open.