Chen–Yang–Yuan–Zhang conjecture on cliques in graphs avoiding disjoint copies of $P_3$

Let $n$, $k$, and $s$ be integers, and let $P_3$ denote the path on three vertices. Write $kP_3$ for the disjoint union of $k$ copies of $P_3$, and let $M_n$ be the graph consisting of $\lfloor n/2\rfloor$ independent edges and at most one isolated vertex. Define

$$f(n,k,s)=\binom{k-1}{s}+(n-k+1)\binom{k-1}{s-1}+\left\lfloor\frac{n-k+1}{2}\right\rfloor\binom{k-1}{s-2}.$$

For a graph $G$, let $\operatorname{ex}(n,K_s,kP_3)$ denote the maximum number of copies of $K_s$ in an $n$-vertex graph containing no subgraph consisting of $k$ vertex-disjoint copies of $P_3$. Let $I_{n-3k+1}$ be an independent graph on $n-3k+1$ vertices. Chen–Yang–Yuan–Zhang's conjecture. If $n\geq 3k$ and $s\geq 3$, then

$$\operatorname{ex}(n,K_s,kP_3)=\max\left\{\binom{3k-1}{s},f(n,k,s)\right\}.$$

Moreover, every extremal graph $G$ satisfies the following conditions: for $3\leq s\leq k+1$, either $G$ is a subgraph of $K_{k-1}+M_{n-k+1}$, or

$$K_{3k-1}\cup I_{n-3k+1}\subseteq G\subseteq K_{3k-1}\cup M_{n-3k+1};$$

for $k+2\leq s\leq 3k-1$,

$$K_{3k-1}\cup I_{n-3k+1}\subseteq G\subseteq K_{3k-1}\cup M_{n-3k+1}.$$

Here $+$ denotes disjoint union with all edges between the summands, and $I_{n-3k+1}$ has no edges. The conjecture gives an exact extremal formula together with a classification of all extremal graphs; its status is not resolved in the supplied source context.