Ma and Yuan's clique-cycle conjecture

Let $G$ be a 2-connected $n$-vertex graph with $n\ge 3$, and let $uv$ be an edge of $G$. Let $k\ge 4$ and $s\ge 2$ be integers, and write

$$n-2=r(k-3)+t,$$

where $0\le t\le k-4$. Here $N_s(G)$ denotes the number of copies of $K_s$ in $G$. Ma and Yuan's conjecture. If

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

then $G$ contains a cycle on at least $k$ vertices containing the edge $uv$.

This conjecture is the clique analogue of Fan's theorem, which gives the corresponding sharp edge bound for $s=2$. Ma and Yuan indicated that it would be a key tool for proving a more general stability result; its resolution is not given in the source.