The prime-cycle clique extremal conjecture

Let $p$ be a prime number, and let $\mathcal C_{\geq p}^{\mathrm{prime}}$ be the family of all cycles of length at least $p$. For integers $n$ and $r$ satisfying $r<p$, let $ex(n,K_r,\mathcal C_{\geq p}^{\mathrm{prime}})$ denote the maximum number of copies of $K_r$ in an $n$-vertex graph containing no member of $\mathcal C_{\geq p}^{\mathrm{prime}}$. Prime-cycle clique extremal conjecture.

$$ex(n,K_r,\mathcal C_{\geq p}^{\mathrm{prime}})\leq \frac{n-1}{p-2}\binom{p-1}{r}.$$

Equality holds if and only if $n-1$ is divisible by $p-2$ and the extremal graph is a connected $n$-vertex graph consisting of $\frac{n-1}{p-2}$ maximal $2$-connected blocks, each isomorphic to $K_{p-1}$. This proposes a prime-parameter analogue of the preceding clique bound for graphs with forbidden long cycles; no resolution is supplied in the source.