Counting triangles in graphs without suspensions of paths

For a graph $H$, the suspension $\widehat H$ of $H$ is a graph obtained from $H$ by joining a new vertex to every vertex of $H$. The generalized Turán number $\textrm{ex}(n,K_3,\widehat H)$ is the maximum number of triangles in an $n$-vertex $\widehat H$-free graph. For fixed $k \ge 4$, Mubayi and M