Chen–Yang–Yuan–Zhang conjecture for generalized Turán numbers of linear paths

For integers $n$, $k$, and $s$, let $H(n,k,q)$ be the graph construction defined above, and for $s\geq 2$ write

$$h_s(n,k,q):=N\bigl(K_s,H(n,k,q)\bigr).$$

Let $\operatorname{ex}(n,K_s,kP_3)$ denote the maximum number of copies of $K_s$ in an $n$-vertex graph containing no $k$ vertex-disjoint copies of the path $P_3$.

Chen–Yang–Yuan–Zhang conjecture. For $n\geq 3k$ and $s\geq 3$,

$$\operatorname{ex}(n,K_s,kP_3)=\max\{h_s(n,k,0),h_s(n,k,k-1)\}.$$

Moreover, if $3\leq s\leq 3k-1$, then every extremal graph $G$ satisfies either

$$G\subseteq H(n,k,k-1),$$

or

$$K_{3k-1}\cup I_{n-3k+1}\subseteq G\subseteq H(n,k,0).$$

This generalizes the known exact edge-counting result for $P_3$-packings to clique counts. The conjecture compares the two constructions that are extremal when $s=2$; the asserted characterization further restricts all extremal graphs for $3\leq s\leq 3k-1$.