Linear-exponential growth conjecture for off-diagonal hypergraph Ramsey numbers

For a fixed $3$-graph $H$, write $r(H,K_n^{(3)})$ for the least $N$ such that every red-blue coloring of the edges of $K_N^{(3)}$ contains a red copy of $H$ or a blue copy of $K_n^{(3)}$. Linear-exponential growth conjecture. There exists a $3$-graph $H$ such that

$$r(H,K_n^{(3)})=2^{\Theta_H(n)}.$$

The source notes that no such single fixed $3$-graph was known, although this growth rate occurs for suitable families of forbidden $3$-graphs. The conjecture remains open.