Egecioglu–Redmond–Ryavec 3-conjecture for a four-term polynomial recurrence

Let $\{p_n(z)\}_{n\in\mathbb N}$ be defined by

$$p_n(z)=zp_{n-1}(z)-Cp_{n-2}(z)-p_{n-3}(z),$$

where $p_{-2}(z)=p_{-1}(z)=0$, $p_0(z)=1$, and $C\in\mathbb R$. Egecioglu–Redmond–Ryavec 3-conjecture. All polynomials in the sequence have only real zeros if and only if $C\geq 3$; moreover, if $C>3$, the zeros of $p_{n+1}(z)$ and $p_n(z)$ interlace for every $n\in\mathbb N$. This conjecture arose in enumerative combinatorics, including problems on alternating sign matrices with vertical symmetries and nonintersecting lattice paths. The supplied text does not establish its resolution status.