Zero-free disk conjecture for matching moment generating functions

Let $F_{k,m,n}(t)=\mathbb{E}(\exp(t\cdot C_{k,m,n}))$ be the moment generating function of the minimum cost $C_{k,m,n}$ of a $k$-matching in the random bipartite matching model. Zero-free disk conjecture. The moment generating function $F_{k,m,n}(t)$ has no complex zeros in the open disk of radius $mn/k$ centered at the origin. The point $t=mn/k$ is the location of the first pole of $F_{k,m,n}(t)$, so the conjecture is equivalently that the Taylor series for $F_{k,m,n}(t)$ and $\log F_{k,m,n}(t)$ have the same radius of convergence. The source presents this as a conjecture supported by numerical computations; no proof or disproof is given.