Fox–Lee–Sudakov lower bound for topological cliques

Let $G$ be a graph, let $\chi(G)$ denote its chromatic number, and let $\sigma(G)$ be the largest integer $t$ such that $G$ contains a subdivision $TK_t$.

Fox–Lee–Sudakov conjecture. There is a constant $c>0$ such that every graph $G$ with $\chi(G)=k$ satisfies

$$\sigma(G)\ge c\sqrt{k\log k}.$$

The conjecture gives a quantitative lower bound on the size of a topological clique in terms of chromatic number. The supplied source gives no evidence of resolution, so it remains open.