The $C_{3\ell+1}$-free clique extremal conjecture

Let $k\geq 2$ and let $n$ be sufficiently large. Let $G$ be an $n$-vertex graph containing no cycle $C_{3\ell+1}$ for any integer $\ell\geq k$. For $r\geq 2$, write $N_r(G)$ for the number of cliques of size $r$ in $G$. Clique extremal conjecture.

$$N_r(G)\leq \frac{n-1}{3k-1}\binom{3k}{r}.$$

Equality holds if and only if $n-1$ is divisible by $3k-1$ and $G$ is a connected $n$-vertex graph consisting of $\frac{n-1}{3k-1}$ maximal $2$-connected blocks, each isomorphic to $K_{3k}$. This conjecture asks whether the sharp block-decomposition phenomenon proved in the paper for sufficiently restricted odd circumferences extends to the family of forbidden cycle lengths congruent to $1$ modulo $3$; its resolution is not supplied here.