Numerical threshold conjecture for cubic overpartition Jensen polynomials

Let $\overline{a}(n)$ denote the cubic overpartition function, and define its Jensen polynomials by

$$J_{\overline{a}}^{d,n}(X):=\sum_{i=0}^{d}\binom{d}{i}\overline{a}(n+i)X^i.$$

For each degree $d$, let $N_{\overline{a}}(d)$ be the minimum integer such that $J_{\overline{a}}^{d,n-1}(X)$ is hyperbolic for all $n\geq N_{\overline{a}}(d)$. The numerical threshold conjecture. The observed threshold values are

\begin{array}{c|ccccc} d&3&4&5&6&7\hline N_{\overline{a}}(d)&39&89&172&279&423 \end{array}

These values are based on numerical evidence from Mathematica; the source does not provide a proof or resolution of the conjecture.