Mubayi–Stein threshold conjecture for tight-path Ramsey parameters

For integers $3\leq t\leq k$, let $\overline r_k(k+1,t;P_n)$ be the least $N$ such that every red/blue coloring of the $k$-sets of an $N$-vertex set contains a blue tight path $P_n$ or has a $(k+1)$-vertex set inducing at least $t$ red edges. Define $\operatorname{twr}_1(x)=x$ and $\operatorname{twr}_{i+1}(x)=2^{\operatorname{twr}_i(x)}$. Mubayi–Stein conjecture. There are positive constants $c=c(k,t)$ and $c'=c'(k,t)$ such that

$$\operatorname{twr}_{t-2}(n^c)<\overline r_k(k+1,t;P_n)<\operatorname{twr}_{t-2}(n^{c'}).$$

This is the tight-path analogue of the Erdős–Hajnal tower-growth conjecture for cliques. The source presents it as open.