Polynomial bound conjecture for 3-uniform clique versus monotone-path Ramsey numbers

Let $K_s$ be the complete $3$-uniform hypergraph on $s$ vertices, and let $P_n$ be the ordered monotone path on $n$ vertices. Write $r_3(K_s,P_n)$ for the least $N$ such that every red-blue coloring of the triples of an $N$-vertex ordered set contains a red copy of $K_s$ or a blue copy of $P_n$.

Polynomial bound conjecture. We have

$$r_3(K_s,P_n)=O(n^c),$$

where $c=c(s)$.

The conjecture concerns the growth of the Ramsey number when $s$ is fixed and $n$ tends to infinity. The paper proves a quasipolynomial upper bound for this quantity, while the conjectured polynomial bound remains open.