The edge-containing long-cycle clique conjecture

Let $G$ be a $2$-connected graph on $n$ vertices, and let $ab$ be an edge of $G$. Let $r\geq 4$ and $s\geq 2$ be integers, and write

$$n-2=x(r-3)+t$$

for some $0\leq t\leq r-4$. Here $N_s(G)$ denotes the number of copies of $K_s$ in $G$. Edge-containing long-cycle conjecture. If

$$N_s(G)>x\binom{r-1}{s}+\binom{t+2}{s},$$

then $G$ contains a cycle on at least $r$ vertices that contains the edge $ab$.

This conjecture would strengthen the paper's main stability theorem by extending it to all ranges of $n$, and predicts that exceeding the stated extremal clique count forces every prescribed edge of a 2-connected graph to lie on a sufficiently long cycle.