Zariski density conjecture for homogeneous polynomials on the rock-paper-scissors algebra

Let $\tilde{\mathfrak M}$ be the rock-paper-scissors algebra with unit over a field $\mathbb F$, and let $p$ be a homogeneous polynomial in one variable whose sum of coefficients is nonzero. Zariski density conjecture. The image set of $p$ evaluated on $\tilde{\mathfrak M}$ is Zariski dense in $\tilde{\mathfrak M}$. This conjecture concerns the unresolved behavior of semi-homogeneous polynomial evaluations; the surrounding discussion indicates that the section contains more questions than answers, and no resolution is supplied here.