Constant-cycle subvariety vanishing conjecture for moduli spaces of sheaves on K3 surfaces

Let $X$ be a K3 surface and ${\mathsf M}$ a moduli space of stable sheaves on $X$ of dimension $2n$. Let $V\subset {\mathsf M}$ be a constant-cycle subvariety, meaning that all points of $V$ have the same class in $CH_0({\mathsf M})$. Let $\overline{\Delta}_{0j}$ denote the modified diagonal correspondence between the first ${\mathsf M}$ factor and the $j$th factor of ${\mathsf M}^{n+1}$. Constant-cycle subvariety vanishing conjecture. One has

$$[V]\cdot\overline{\Delta}_{01}\cdot\overline{\Delta}_{02}\cdots\overline{\Delta}_{0,n+1}=0$$

in $CH_{\star}({\mathsf M}\times {\mathsf M}^{n+1})$, where $[V]$ is pulled back from the first ${\mathsf M}$ factor. This generalizes the corresponding zero-cycle identity and seeks a uniform Chow-theoretic relation for constant-cycle subvarieties of all dimensions.