Lassalle's Kerov–Lassalle positivity conjecture for cumulants

Let $cgamma=-\delta$, let $cR_2,cR_3,\cdots$ be the free cumulants, and let $ckappa_{\bullet}$ denote the cumulants of Jack characters. For positive integers $k_1,\cdots,k_\ell$, consider the signed cumulant

$$(-1)^{\ell-1}\kappa_{\bullet}(\operatorname{Ch}_{k_1},\dots,\operatorname{Ch}_{k_\ell}).$$

Kerov–Lassalle cumulant-positivity conjecture. For all integers $k_1,\dots,k_\ell\geq 1$, the polynomial which expresses this signed cumulant in terms of $cgamma,\mathcal{R}_2,\mathcal{R}_3,\dots$ has non-negative integer coefficients. This extends the single-character Kerov–Lassalle positivity conjecture to cumulants; explicit examples support it, but the source does not give a resolution.