  For any real numbers $a,\ b$, and $c$, we form the sequence of polynomials
$\{P_n(z)\}_{n=0}^\infty$ satisfying the four-term recurrence \[
P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in\mathbb{N}, \] with the
initial conditions $P_0(z)=1$ and $P_{-n}(z)=0$. We find necessary and
sufficient conditions on $a,\ b$, and $c$ under which the zeros of $P_n(z)$ are
real for all $n$, and provide an explicit real interval on which
$\displaystyle\bigcup_{n=0}^\infty\mathcal{Z}(P_n)$ is dense, where
$\mathcal{Z}(P_n)$ is the set of zeros of $P_n(z)$.
