The clique-density conjecture for complements of string graphs

Let $s\ge 1$, let $\varepsilon>0$, and let $G$ be the complement of a string graph. Here $\varrho_{4s+1}(G)$ denotes the density of copies of $K_{4s+1}$ in $G$, and $K_{s+1}[t]$ denotes the complete $(s+1)$-partite graph with parts of size $t$. The string-graph clique-density conjecture. If

$$\varrho_{4s+1}(G)>0,$$

then $G$ contains a copy of $K_{s+1}[\delta n]$, where $\delta=\delta(\varepsilon)>0$. This proposes replacing the positive edge-density hypothesis in the known string-graph result by positive clique density; the conjecture remains open.