Positivity conjecture for cumulant coefficients of Jack characters

Let $cpi_1,\cdots,cpi_\ell$ be partitions, and define their Jack-character cumulant expansion by

$$\kappa_{\bullet}(\operatorname{Ch}_{\pi_1},\dots,\operatorname{Ch}_{\pi_\ell})=\sum_\sigma d^\sigma_{\pi_1,\dots,\pi_\ell}(\delta)\operatorname{Ch}_\sigma.$$

Cumulant-coefficient positivity conjecture. For partitions $cpi_1,\dots,cpi_\ell$, the polynomial $(-1)^{\ell-1}d^\sigma_{\pi_1,\dots,\pi_\ell}(cdelta)$ has non-negative integer coefficients. This generalizes the structure-coefficient positivity conjecture, which is the case $cell=2$ in the formulation discussed by the source; no resolution is stated.