Lv et al.'s generalized Turán conjecture for triangles and expanded paths and cycles

Let $P_k^{\triangle}$ and $C_k^{\triangle}$ denote the graphs obtained by the relevant expansion of the path $P_k$ and cycle $C_k$, respectively. For graphs $G_1$ and $G_2$, their join G_1\mathbin{\;{\rotatebox{90}{\textnormal{\small\Bowtie}}}}G_2 has vertex set $V(G_1)\cup V(G_2)$ and contains all edges of $G_1$ and $G_2$, together with every edge between $V(G_1)$ and $V(G_2)$. Let $T(n)$ be a balanced complete bipartite graph, let $X,Y$ be its parts with $|X|\le |Y|$, and let $T^+(n)$ be obtained from $T(n)$ by adding an edge inside $X$. For a positive integer $t$, define

$$S(n,t):=K_t\mathbin{\;{\rotatebox{90}{\textnormal{$\small\BowTie$}}}}T(n-t),\qquad S^+(n,t):=K_t\mathbin{\;{\rotatebox{90}{\textnormal{$\small\BowTie$}}}}T^+(n-t).$$

For graphs $Q$ and $F$, let $N(Q,F)$ be the number of copies of $Q$ in $F$, and let $\mathrm{ex}(n,Q,F)$ be the maximum number of copies of $Q$ in an $n$-vertex $F$-free graph.

Lv et al.'s conjecture. For $k\ge 4$ and sufficiently large $n$,

$$\mathrm{ex}(n,K_3,P_k^{\triangle})=\mathrm{ex}(n,K_3,C_k^{\triangle})= \begin{cases} N\left(K_3,S(n,\left\lfloor \frac{k-1}{2}\right\rfloor)\right),&\text{if $k$ is odd},\\ N\left(K_3,S^+(n,\left\lfloor \frac{k-1}{2}\right\rfloor)\right),&\text{if $k$ is even}. \end{cases}$$

This extends exact generalized Turán results for several expanded paths and cycles and predicts the extremal triangle counts for all $k\ge 4$ once $n$ is sufficiently large; its resolution is not established in the supplied context.