Weinberger's contractibility conjecture for preferential attachment clique complexes

Let $X(\infty,\delta,m(\cdot))$ be the clique complex of the graph obtained by adding one node and $m(t)$ preferential-attachment edges at each time $t$, where $m:\mathbb{Z}_0^+\to\mathbb{Z}_0^+$ is slowly growing. Weinberger's contractibility conjecture. The complex

$$X(\infty,\delta,m(\cdot))$$

is contractible. This conjecture concerns whether all holes are eventually filled in a preferential attachment process whose number of newly added edges grows slowly; the source attributes it to Weinberger and gives no resolution.