Binary path configuration conjecture for avalanche homology

Let $P_n$ be the directed path on $n$ vertices, let $\beta_i$ denote the $i$-th Betti number of its avalanche complex, and let a binary configuration be an initial configuration whose entries are all either $0$ or $1$. Binary path homology conjecture. If $\mathbf{c}_0$ is a binary configuration, then for $\mathcal{A}(P_n,\mathbf{c}_0)$ we have

$$\beta_i=0 \quad\text{for all } i>\left\lfloor\frac{n}{2}\right\rfloor-1.$$

This would identify the configuration in Proposition 1 as having the highest possible degree of homology among binary initial configurations. The conjecture is refuted once the binary condition is relaxed: $\mathcal{A}(P_7,(5,0,1,1,1,1,1))$ has $\beta_4=1$.