The family-level tower lower-bound conjecture for tightly connected hypergraphs

For integers $k$ and $s$, let $\mathcal{H}_s^{(k)}$ be the family of $s$-tightly connected $k$-graphs that are not $k$-partite. A hypergraph is $s$-tightly connected if any two edges can be joined by a sequence of edges in which consecutive edges share at least $s$ vertices. 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 family-level tower lower-bound conjecture. If $k>s$, then there exists a positive constant $c$ such that

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

This is explicitly proposed as a stronger version of the corresponding conjecture for each individual tightly connected hypergraph. It remains open.