Erdős–Hajnal tower-growth conjecture for $r_k(k+1,t;n)$

For integers $k\geq3$ and $2\leq t\leq k$, let $r_k(k+1,t;n)$ be the Erdős–Hajnal Ramsey parameter. Define $\operatorname{twr}_1(x)=x$ and $\operatorname{twr}_{i+1}(x)=2^{\operatorname{twr}_i(x)}$. Erdős–Hajnal conjecture. There is a constant $c=c(k,t)>0$ such that

$$r_k(k+1,t;n)\geq \operatorname{twr}_{t-1}(c n).$$

The source gives the matching tower-level upper bound $r_k(k+1,t;n)\leq\operatorname{twr}_{t-1}(n^c)$ and conjectures that this is the correct tower growth rate. The cases $t=2$ and partial bounds for $t=3$ are known, but the general lower bound remains open.