Is it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\tau(G)=o_c(n)$?

Similarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\tau(G)<(1-c)n$.

OPEN

For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).

Is it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\tau(G)=o_c(n)$?

Similarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\tau(G)<(1-c)n$.

A problem of Erdős, Gallai, and Tuza [EGT92], who proved for the latter question that $k_c(n) \geq n^{c'/\log\log n}$ for some $c'>0$, and that if every clique has size least $k$ then $\tau(G) \leq n-(kn)^{1/2}$. Bollobás and Erdős proved that if every maximal clique has at least $n+3-2\sqrt{n}$ vertices then $\tau(G)=1$ (and this threshold is best possible).

See also [610] and the entry in the graphs problem collection.