The $m$-shuffle conjecture

Fix positive integers $m,n$ and set $\delta_n=(n-1,n-2,\ldots,1,0)$. For a partition $\lambda\subseteq m\delta_n$, let $T$ be a semistandard tableau of shape $(\lambda+(1^n))/\lambda$, and let $\operatorname{dinv}_m(T)$ be the generalized d-inversion statistic defined using $d_m(i,j)=mi+j$. Define

$$D_n^{(m)}(z;q,t)=\sum_{\lambda\subseteq m\delta_n}\;\sum_{T\in\operatorname{SSYT}(\lambda+(1^n)/\lambda)}t^{|m\delta_n/\lambda|}q^{\operatorname{dinv}_m(T)}z^T.$$

Here $\nabla^m$ is the $m$th power of the nabla operator and $e_n,h_\mu$ have their usual symmetric-function meanings. The $m$-shuffle conjecture. One has

$$\nabla^m e_n(z)=D_n^{(m)}(z;q,t).$$

Equivalently, for every partition $\mu$,

$$\langle\nabla^m e_n,h_\mu\rangle=\sum_{\lambda\subseteq m\delta_n}\;\sum_{T\in\operatorname{SSYT}(\lambda+(1^n)/\lambda,\mu)}t^{|m\delta_n/\lambda|}q^{\operatorname{dinv}_m(T)}.$$

This extends the $m=1$ tableau formula and is related to conjectures on generalized diagonal coinvariants. The supplied text gives no resolution.