The graded character conjecture for derivative spaces $D_{\mu}$

Let $\mu$ be a partition of $n$, and let $D_{\mu}=\bigoplus_{r,s}(D_{\mu})_{r,s}$ be the doubly graded $S_n$-module of partial derivatives of $\Delta_{\mu}$. Write $\operatorname{ch}V$ for the character of an $S_n$-module $V$, and let $\chi^{\lambda}$ be the irreducible character indexed by the partition $\lambda$ of $n$.

Graded character conjecture. We have

$$\tilde{K}_{\lambda\mu}(q,t)=\sum_{r,s}t^rq^s\left\langle\chi^{\lambda},\operatorname{ch}(D_{\mu})_{r,s}\right\rangle.$$

This stronger conjecture implies Macdonald positivity by expressing each coefficient as a graded multiplicity. The source presents it as a conjecture related to the $n!$ conjecture, without supplying a resolution status here.