Katona–Xiao conjecture for the path-and-clique Turán problem

Let $P_k$ be a path on $k$ vertices, let $K_m$ be the complete graph on $m$ vertices, and let $T(n,r)$ denote the Turán graph on $n$ vertices with independence number at most $r$. Write $I_s$ for the edgeless graph on $s$ vertices, $\delta_k$ for the relevant path parameter, and $G\vee H$ for the join of graphs $G$ and $H$. Assume $m+1\leq k\leq 2m-1$.

Katona–Xiao conjecture. If $k$ is odd and $(k-1)\mid n$, then the disjoint union of $n/(k-1)$ copies of $T(k-1,m-1)$ gives the maximum number of edges in a graph containing neither $P_k$ nor $K_m$, while for even $k$, $T(\delta_k,m-2)\vee I_{n-\delta_k}$ is extremal for sufficiently large $n$.

This conjecture concerns the remaining range $m<k\leq 2m-1$ in the generalized Turán problem forbidding both a path and a clique. The corresponding connected and unrestricted extremal values are known when $k>2m-1$, but this intermediate range remains unresolved.