Erdős–Hajnal off-diagonal hypergraph Ramsey conjecture

For integers $4\leq k<s$ and $n$, let $r_k(s,n)$ be the least $N$ such that every red/blue coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a red copy of the complete $k$-uniform hypergraph on $s$ vertices or a blue copy on $n$ vertices. Define $\operatorname{twr}_1(x)=x$ and $\operatorname{twr}_{i+1}(x)=2^{\operatorname{twr}_i(x)}$. Erdős–Hajnal conjecture. There are constants $c,c'>0$ such that

$$\operatorname{twr}_{k-1}(cn)<r_k(s,n)<\operatorname{twr}_{k-1}(c'n).$$

This conjecture predicts the correct tower growth for the off-diagonal numbers with smaller values of $s$ than those covered by the known lower bound. It is verified in the source for $s\geq k+3$, while the cases $r_4(5,n)$ and $r_4(6,n)$ are substantially harder.