The cyclic quandle cohomology suspension conjecture for $R_p$

Let $p$ be an odd prime, let $R_p$ be the dihedral quandle of order $p$, and let $\sigma^*$ denote the cohomology homomorphism induced by the quandle chain map $\sigma$. Using the identifications

$$H_Q^3(R_p;\mathbb{Z}_p)\cong\mathbb{Z}_p,\qquad H_Q^4(R_p;\mathbb{Z}_p)\cong\mathbb{Z}_p^2,$$

consider the induced map $\sigma^*$.

The cyclic quandle cohomology suspension conjecture. The map

$$\sigma^* \colon H_Q^3(R_p;\mathbb{Z}_p) \to H_Q^4(R_p;\mathbb{Z}_p)$$

is an injective homomorphism $\mathbb{Z}_p\to\mathbb{Z}_p^2$ such that $1\mapsto(-1,0)$.

The preceding proposition establishes the same map with the second coordinate known only to be some $n\in\mathbb{Z}_p$; computer calculations verify $n=0$ for $p=3,5,7,11,13$. The conjecture asserts that this vanishing holds for every odd prime $p$.