Positivity conjecture for cumulants of minimum matching costs

Let $C_{k,m,n}$ be the minimum cost of a $k$-matching in the random bipartite matching model, and let its cumulants $\kappa_p$ be defined by

$$\log F_{k,m,n}(t)=\sum_{p=1}^{\infty}\frac{\kappa_p t^p}{p!},$$

where $F_{k,m,n}(t)=\mathbb{E}(\exp(t\cdot C_{k,m,n}))$ is the moment generating function. Positivity conjecture. All the cumulants of $C_{k,m,n}$ are positive. This conjecture is based on computations for all $k\leq m\leq n\leq 15$ and less systematic computations for larger parameters; no proof or disproof is given in the source.