Diameter-dominance conjecture for Rips complexes of preferential attachment graphs

Let $G(T,\delta,m)$ be the preferential attachment graph and let its Rips complex of radius $r$ be formed using the graph metric. Diameter-dominance conjecture. The Betti numbers of this Rips complex are dominated by cycles whose diameter is $r+1$. The conjecture is motivated by the intuition that larger cycles arise less frequently; the supplied text gives no resolution.