Lassalle's Kerov–Lassalle positivity conjecture for single Jack characters

Let $cgamma=-\delta$, and let $cR_2,cR_3,\cdots$ denote the free cumulants, viewed as functions on Young diagrams. For each partition $cpi$, the Jack character $coperatorname{Ch}_\pi$ is expressed by a Kerov–Lassalle polynomial in $cgamma,cR_2,cR_3,\cdots$. Kerov–Lassalle positivity conjecture. For each integer $k\geq 1$, the polynomial which expresses the Jack character $coperatorname{Ch}_k$ in terms of the variables $cgamma,\mathcal{R}_2,\mathcal{R}_3,\dots$ has non-negative integer coefficients. This is Lassalle's positivity conjecture for one-part Jack characters, motivated by explicit formulas; the source does not state that it has been resolved.