Existence of extremal graphs for clique density

Let $\mathcal{G}(\Delta,\omega)$ be the class of graphs with maximum degree at most $\Delta$ and clique number at most $\omega$, and let $f_t(\Delta,\omega)$ be the supremum of $\rho_t(G)$ over this class. Existence conjecture. For all $t\geq 3$ and $3\leq\omega\leq\Delta+1$, there exists a graph $G\in\mathcal{G}(\Delta,\omega)$ such that

$$\rho_t(G)=f_t(\Delta,\omega).$$

The paper notes that its results do not establish attainment of the supremum for all relevant parameters; this conjecture asks for an extremal graph in every allowed case and remains open.