Kahle's bouquet of spheres conjecture for random clique complexes

Let $d$ be an integer with $d\geq 3$, let $\alpha$ satisfy

$$\frac{1}{d+1}<\alpha<\frac{1}{d},$$

and let $\Delta\sim\Delta(n,n^{-\alpha})$ be a random clique complex. Kahle's bouquet of spheres conjecture. With high probability, $\Delta$ is homotopy equivalent to a bouquet of $d$-dimensional spheres. This would strengthen the known rational homology-vanishing and nonvanishing results in the stated probability range by determining the homotopy type; the conjecture remains open in the source.