Triangle extremality conjecture for the graph $\operatorname{BT}(3)$

For $k\geq 2$, let $\widetilde{\operatorname{BT}}(k)$ be obtained from $T_2(2k)$ by deleting an edge and adjoining a new vertex to the two endpoints of that edge, and let

$$\operatorname{BT}(k)=\widetilde{\operatorname{BT}}(k)\vee I_{k+1}.$$

Let $f_3(\Delta,\omega)$ be the supremum of the triangle density $\rho_3$ among graphs with maximum degree at most $\Delta$ and clique number at most $\omega$. Triangle extremality conjecture. For $k=3$,

$$f_3(2k+1,3)=\rho_3(\operatorname{BT}(k)).$$

The preceding theorem proves that $\operatorname{BT}(k)$ beats the relevant Turán graph for $k=2,3$; the conjecture asserts its optimality in the case $k=3$, namely for parameters $(\Delta,\omega)=(7,3)$.