The gamma lower-bound conjecture for weak saturation of hypergraphs

Let $r\ge 2$, $s\ge 2$, and $k\ge r+1$ be integers. Let

$$\delta=\binom{k-s+1}{r-s+1}.$$

Let $H$ be an $r$-uniform hypergraph with $s(H)=s$ and $\delta_{s-1}(H)=\delta$. The gamma lower-bound conjecture. Then

$$\gamma_{s,H}\ge \frac{\delta}{\binom{r}{s-1}}-\frac{1}{\binom{k}{s-1}}.$$

This would sharpen the general lower bounds for weak saturation in the specified parameter regime; the source suggests proving it using methods analogous to those used for the existing lower bound on $\gamma_{s,H}$.