Mubayi–Mukherjee conjecture on triangles avoiding a suspended path

Let $n\geq k\geq 4$. For a graph $F$, let $\widehat{F}$ denote the graph obtained by adding a vertex adjacent to every vertex of $F$, and let $\operatorname{ex}(n,H,F)$ be the maximum number of copies of $H$ in an $n$-vertex graph containing no copy of $F$. Let $P_k$ be the path on $k$ vertices. Mubayi–Mukherjee conjecture.

$$\operatorname{ex}(n,K_3,\widehat{P_k})=\left\lfloor \frac{k-2}{2}\right\rfloor\cdot\frac{n^2}{8}+o(n^2).$$

The conjecture asserts that the lower-bound construction of Mubayi and Mukherjee is asymptotically optimal for every fixed $k\geq 4$, improving the currently stated upper bound to the conjectured leading term.