Exponential decay conjecture for the Hamming-cube theta number

Let $H(n,s)$ denote the Hamming hypergraph on $\mathbb{H}^n$ whose edges are $s$-triangles, and for fixed $c>1$ let $s(n,c)$ be the even integer closest to $n/c$. The quantity $mathvartheta(H(n,s(n,c))) / 2^n$ is the normalized theta number for this hypergraph.

Hamming-cube decay conjecture. For every fixed $c>2$, the quantity

$$\vartheta(H(n,s(n,c))) / 2^n$$

decays exponentially fast with $n$, whereas

$$\vartheta(H(n,s(n,2))) / 2^n$$

decays linearly fast with $n$.

This conjecture concerns the asymptotic density bound produced by the theta number for subsets of the Hamming cube that avoid $s$-triangles. The exponential-versus-linear transition is supported by numerical evidence, but no proof or resolution is supplied here.