Mubayi–Stein strengthening for tight-path Ramsey numbers

For fixed $k\geq4$, let $P_{k+1}$ denote the relevant $k$-uniform tight path, and let $\overline r_k(P_{k+1},n)$ be its Ramsey number against a complete $k$-uniform hypergraph on $n$ vertices. Define $\operatorname{twr}_1(x)=x$ and $\operatorname{twr}_{i+1}(x)=2^{\operatorname{twr}_i(x)}$. Mubayi–Stein conjecture.

$$\overline r_k(P_{k+1},n)\geq\operatorname{twr}_{k-1}(\Omega(n)).$$

This strengthens the Erdős–Hajnal prediction and would improve the known lower bound by one tower level. The source relates the case $k=4$ to the diagonal 3-uniform Ramsey conjecture.