Hinz–Młotkowski conjecture on free Poisson double convolution powers

Let $\bm{\pi}$ be the free Poisson distribution with mean $1$, and for $s,t>0$ define the double free convolution power

$$\tilde{\bm{\pi}}_{st}=(\bm{\pi}^{\boxtimes s})^{\boxplus t}.$$

The moments of this measure are given by $\tilde{m}_0(s,t)=1$ and, for $n\geq1$,

$$\tilde{m}_n(s,t)=\sum_{k=1}^n\frac{t^k}{n}\binom{n}{k-1}\binom{ns}{n-k}.$$

Hinz–Młotkowski conjecture. The double power $\tilde{\bm{\pi}}_{st}$ is a probability measure if and only if $\max(s,t)\geq1$. Equivalently, the sequence $(\tilde{m}_n(s,t))_{n\geq0}$ is positive definite if and only if $\max(s,t)\geq1$.