Exponential upper-bound conjecture for the three-uniform color-avoiding Ramsey function

Let $R_3(n;3,2)$ be the least integer $N$ such that every three-coloring of the edges of the complete three-uniform hypergraph $K_N^{(3)}$ contains a copy of $K_n^{(3)}$ using at most two colors. Exponential upper-bound conjecture. There exists a constant $c>0$ such that

$$R_3(n;3,2)\leq 2^{n^c}.$$

The corresponding color-avoiding problem is open, and even in uniformity three the quantitative behavior for $r=3$ and $s=2$ is poorly understood. Existing methods give an exponential-type lower bound and a double-exponential upper bound, so the conjecture would substantially improve the known upper bound and determine the tower height.