Monomial positivity conjecture for the Jack super nabla operator on elementary functions

Let $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define the coefficient $f^\pi_{\mu,\nu}(\alpha)$ by

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

where $e_\pi$ denotes the elementary symmetric function. Monomial positivity conjecture. For all such $\pi,\mu,\nu$, one has

$$f^\pi_{\mu,\nu}(\alpha)\in\mathbb{N}[\alpha].$$

This is one of the positivity questions for the Jack super nabla operator on symmetric-function bases, tested computationally for $n\leq 9$. The conjecture asks for positivity in the unshifted parameter $\alpha$ and is motivated by analogous positivity phenomena for related bases.