Conjecture on neural ring homomorphisms of prime-support circulant codes

Let $\mathcal{C}$ be a circulant code on $n$ neurons with support $p$. Let $\operatorname{NRH}\{\mathcal{R}_{\mathcal{C}}\}$ denote the set of neural ring homomorphisms associated with $\mathcal{C}$. Conjecture on prime-support circulant codes. If $p>2$ is prime and $p$ divides $n$, then

$$|\operatorname{NRH}\{\mathcal{R}_{\mathcal{C}}\}|=3n+p^2\left(\dfrac{n}{p}\right)!+p(p+1)n.$$

This conjecture extrapolates a pattern from the preceding theorems for circulant codes; the source gives no resolution.