Linear-exponent conjecture for 3-uniform Ramsey numbers

Let $g_3(s)$ denote the maximum number of edges in an $s$-vertex iterated blowup of a $3$-uniform edge, and let $r_3(s,e;t)$ be the corresponding hypergraph Ramsey number. Conjecture. For all $s>3$,

$$r_3(s,g_3(s)+1;t)=2^{\Omega(t)}.$$

This would sharpen the paper's $t^{2/3}$ exponent in the exponential lower bound and is stated as the expected optimal power of $t$ in the exponent. It remains open.