CFG+Y characterization of polynomial 3-uniform Ramsey growth

Let $H$ be a $3$-uniform hypergraph, let $K_n^3$ be the complete $3$-uniform hypergraph on $n$ vertices, and let $r(H,K_n^3)$ be the corresponding Ramsey number. Call $H$ iterated tripartite if it belongs to the iterated-tripartite class described in the source. CFG+Y conjecture. For any $3$-graph $H$, there is a constant $c$ such that

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

for all $n$ if and only if $H$ is iterated tripartite. The source describes this as an adjacent open problem and as a strengthening of the paper's main theorem.