The tower lower-bound conjecture for tightly connected hypergraph Ramsey numbers

For integers $k$ and $s$, a $k$-graph is a $k$-uniform hypergraph. It is $s$-tightly connected if any two edges $e$ and $f$ are joined by a sequence of edges $e=e_0,e_1,\dots,e_t=f$ such that consecutive edges share at least $s$ vertices. A $k$-graph is $k$-partite if its vertices can be partitioned into $k$ parts so that every edge has one vertex in each part. Define the tower function by

$$t_1(x)=x,\qquad t_i(x)=2^{t_{i-1}(x)}\quad\text{for all }i\geq 2.$$

The tower lower-bound conjecture. If $k>s$ and $H$ is an $s$-tightly connected $k$-graph that is not $k$-partite, then there exists a positive constant $c$ such that

$$r(H,K_n^{(k)})\geq t_s(n^c).$$

This would extend the known exponential lower bound for 2-tightly connected non-$k$-partite hypergraphs to higher levels of the tower. The conjecture is presented as likely difficult, and remains open.