The iterated-tripartite classification conjecture for off-diagonal 3-graph Ramsey numbers

A 3-graph is a 3-uniform hypergraph, and $r(H,K_n^{(3)})$ denotes the smallest $N$ such that every red/blue-coloring of the edges of the complete 3-graph $K_N^{(3)}$ contains either a red copy of $H$ or a blue copy of $K_n^{(3)}$. An iterated blowup of an edge is a complete tripartite 3-graph, or a 3-graph formed by repeatedly placing another iterated blowup in one or more parts; a 3-graph is iterated tripartite when it is contained in such a blowup.

The iterated-tripartite classification conjecture. For a 3-graph $H$, there exists a constant $c$ depending only on $H$ such that

$$r(H,K_n^{(3)})\leq n^c$$

for all $n$ if and only if $H$ is a subgraph of an iterated blowup of an edge.

The forward direction would classify exactly which 3-graphs have polynomial off-diagonal Ramsey numbers. The polynomial upper bound for iterated tripartite 3-graphs was already known, while the converse remains open.