Limit conjecture for the largest root r_{n,m}

For $n,m\in\mathbb N$, let $r_{n,m}$ denote the largest real root of the polynomial associated with the Bergman kernel of the Fock-Bargmann-Hartogs domain $D_{n,m}$. Limit conjecture. For every fixed $m\in\mathbb N$,

$$\lim_{n\to\infty}r_{n,m}=0.$$

The paper proves the corresponding limit as $m\to\infty$ with $n$ fixed, but does not determine the limit in $n$; the displayed claim is supported only by numerical computation and remains open.