Asymptotic-limit conjecture for expected Betti numbers of preferential attachment clique complexes

For $q\geq 2$ and $m\geq 2q$, let $X(T,\delta,m)$ denote the affine preferential attachment clique complex and let $\beta_q(X(T,\delta,m))$ be its $q$-th Betti number. If $1-2q\chi(\delta,m)>0$, the asymptotic-limit conjecture. The limit

$$\lim_{T\to\infty}\frac{E[\beta_q(X(T,\delta,m))]}{T^{1-2q\chi(\delta,m)}}$$

exists. The conjecture seeks a sharper asymptotic description than the estimates established in the paper; the supplied text gives no resolution.