Eventual hyperbolicity conjecture for Jensen polynomials of k-regular partitions

For an integer $k\geq2$, let $p_k(n)$ count the partitions of $n$ having no parts divisible by $k$, so that

$$\sum_{n\geq0}p_k(n)q^n=\prod_{n=1}^{\infty}\frac{1-q^{kn}}{1-q^n}.$$

For integers $d\geq1$ and $n\geq0$, let $J_{p_k}^{d,n}(X)$ be the degree-$d$ Jensen polynomial associated with the sequence $p_k(n)$.

k-regular Jensen polynomial conjecture. Let $d\geq1$ and $k\geq2$ be integers. There exists a positive integer $N(d)$ such that $J_{p_k}^{d,n}(X)$ is hyperbolic for all $n\geq N(d)$.

Here hyperbolic means that all roots are real. The paper states this as an equivalent reformulation of the higher-order Turán claim and then proves the stronger result for $k$-regular partition functions.