Double-exponential lower bound conjecture for r_4(5,n)

Let $r_4(5,n)$ be the off-diagonal Ramsey number for 4-uniform hypergraphs, the least $N$ such that every red-blue coloring of the 4-edges of an $N$-vertex complete hypergraph contains a red copy of the complete 4-uniform hypergraph on 5 vertices or a blue copy on $n$ vertices. The $r_4(5,n)$ lower-bound conjecture. For $n\geq 5$, there is an absolute constant $c>0$ such that

$$r_4(5,n)>2^{2^{n^c}}.$$

The conjecture would improve the best known lower bound for $r_4(5,n)$ from a single exponential of the form $2^{n^{c\log\log n}}$ to a double exponential. The source notes that this conjecture follows from the crucial diagonal conjecture for $r_3(n,n)$ established earlier in the paper's discussion, but it remains unproved here.