Tomon's incomparability-graph blowup conjecture

Let $k\ge 3$, let $\varepsilon>0$, and let $G$ be an $n$-vertex incomparability graph. Here $\varrho_2(G)$ denotes the edge density of $G$, and $K_k[t]$ denotes the complete $k$-partite graph with $k$ parts of size $t$. Tomon's conjecture. If

$$\varrho_2(G)>1-\frac{1}{k-1}+\varepsilon,$$

then $G$ contains a copy of

$$K_k\bigg[\frac{cn}{(\log n)^s}\bigg],$$

where $c=c(\varepsilon,k)>0$ and $s=\lceil\log_2 k\rceil$. This refines the known result with the weaker density threshold $1-\frac{1}{9(k-1)}+\varepsilon$; whether the conjectured threshold guarantees the stated blowup remains open.