  For any real numbers $b,c\in\mathbb{R}$, we form the sequence of polynomials
$\left\{ H_{m}(z)\right\} _{m=0}^{\infty}$ satisfying the four-term recurrence
\[ H_{m}(z)+cH_{m-1}(z)+bH_{m-2}(z)+zH_{m-3}(z)=0,\qquad m\ge3, \] with the
initial conditions $H_{0}(z)=1$, $H_{1}(z)=$$-c$, and $H_{2}(z)=-b+c^{2}$. We
find necessary and sufficient conditions on $b$ and $c$ under which the zeros
of $H_{m}(z)$ are real for all $m$, and provide an explicit real interval on
which ${\displaystyle \bigcup_{m=0}^{\infty}\mathcal{Z}(H_{m})}$ is dense where
$\mathcal{Z}(H_{m})$ is the set of zeros of $H_{m}(z)$.
