Multicoloured Erdős–Hajnal conjecture for graphs

Let $K_k$ and $K_n$ be complete graphs. An $s$-colouring is an edge-colouring using colours in a set of at most $s$ colours; a colouring $d$ of $K_n$ contains a colouring $c_0$ of $K_k$ if there is an injection $\psi:V(K_k)\to V(K_n)$ preserving all edge colours, and it is $c_0$-free otherwise.

Multicoloured Erdős–Hajnal conjecture. For all $1\le s_0\le s$ and $k\ge1$, and every $s_0$-colouring $c_0$ of $K_k$, there exist $\delta,C>0$ such that, for all $n\in\mathbb N$ and every $c_0$-free $s$-colouring $c$ of $K_n$, there are a colour $i\in\{1,\dotsc,s\}$ and a set $X$ of at least $Cn^\delta$ vertices of $K_n$ such that $c(e)\ne i$ for every edge $e$ with both ends in $X$.

This is a multicoloured form of the induced Erdős–Hajnal problem. It is known for some particular forbidden colourings and infinite classes, but is wide open in general.