Exact generalized Turán number for disjoint copies of P3

Determine the exact maximum number of copies of K_s in an n-vertex graph containing no k vertex-disjoint copies of P_3, for all integers n \geq 3k and s \geq 3, by proving that \operatorname{ex}(n,K_s,kP_3)=\max\left\{\binom{3k-1}{s},\,f(n,k,s)\right\}, where 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}. Characterize the extremal graphs according to the stated ranges of s.

Background

The paper studies generalized Turán numbers \operatorname{ex}(n,T,H), which maximize the number of copies of a graph T in an n-vertex H-free graph. The relevant forbidden graph is kP_3, the disjoint union of k paths on three vertices, and the target graph is the clique K_s.

Chen, Yang, Yuan, and Zhang are cited as having proposed the conjecture that the extremal value is the larger of the clique construction K_{3k-1}\cup M_{n-3k+1} and the join construction K_{k-1}+M_{n-k+1}, whose respective clique counts are \binom{3k-1}{s} and f(n,k,s). The paper proves the conjecture for k=2,3; for s\geq k+2; for 3\leq s\leq k when n is sufficiently large; and for s=k+1 when n is sufficiently large. Thus, the conjecture is only partially resolved in the paper, while the displayed assertion gives the broader unresolved problem.