Upper-bound conjecture for the threshold m_0(n)

For $n\in\mathbb N$, let $m_0(n)$ be the least positive integer such that the associated Fock-Bargmann-Hartogs domain $D_{n,m}$ is a Lu Qi-Keng domain, and let $[x]$ denote the nearest integer to $x$. Upper-bound conjecture.

$$m_0(n)\leq [(n+1)\log(n+1)].$$

Moreover, equality holds if and only if $n\leq 10$. This claim is suggested by numerical data and remains open.