Monomial positivity conjecture for the super nabla operator

Fix $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define $d^\pi_{\mu,\nu}(\alpha)$ by

$$\nabla(\boldsymbol{p},\boldsymbol{q})p_\pi=\sum_{\mu,\nu\vdash n}d^\pi_{\mu,\nu}(\alpha)p_\mu m_\nu(\boldsymbol{q}),$$

where $m_\nu$ denotes the monomial symmetric function in the alphabet $\boldsymbol{q}$. Monomial positivity conjecture. The coefficients satisfy $d^\pi_{\mu,\nu}(\alpha)\in\mathbb N[\alpha]$. This conjecture was tested for $n\leq 9$. The Matching–Jack conjecture and power-sum monomial positivity imply positivity in $\mathbb N[b]$; the conjecture proposes positivity in $\mathbb N[\alpha]$ without shifting the parameter.