Generalized Turán extremal graph conjecture for edge blow-ups of cycles and paths

Let $C_k^3$ and $P_k^3$ denote the edge blow-ups of the cycle $C_k$ and path $P_k$, respectively. For integers $p,t$, let $T_p(n-t+1)$ be the balanced $p$-partite complete graph on $n-t+1$ vertices, let

$$H(n,p,t)=K_{t-1}+T_p(n-t+1),$$

and let $H^+(n,p,t)$ be obtained from $H(n,p,t)$ by adding an extra edge within one class of $T_p(n-t+1)$. Write $\operatorname{ex}(n,K_3,F)$ for the maximum number of copies of $K_3$ in an $F$-free graph on $n$ vertices. Generalized Turán extremal graph conjecture. When $k\geq 4$ and $n$ is sufficiently large, $H(n,2,\lfloor\frac{k-1}{2}\rfloor+1)$ is the unique extremal graph for both $\operatorname{ex}(n,K_3,C_k^3)$ and $\operatorname{ex}(n,K_3,P_k^3)$ when $k$ is odd, while $H^+(n,2,\lfloor\frac{k-1}{2}\rfloor+1)$ is the unique extremal graph when $k$ is even. The conjecture extends the paper's results for the edge blow-ups of triangles and three-edge paths to general edge blow-ups of cycles and paths, with the asserted uniqueness expected for sufficiently large $n$.