Hitting-time conjecture for the Erdős–Ko–Rado property in random Kneser graphs

For integers $n$ and $k$, let $K_i(n,k)$ denote the random subgraph process of the Kneser graph, and define

$$\tau_{\alpha}=\min\left\{i:\alpha(K_i(n,k))=\binom{n-1}{k-1}\right\},$$ $$\tau_{\textnormal{super}}=\min\{i:K_i(n,k)\text{ has no independent superstars}\},$$ $$\tau_{\textnormal{near}}=\min\{i:K_i(n,k)\text{ has no independent near-stars}\},$$ $$\tau_{\textnormal{EKR}}=\min\{i:K_i(n,k)\text{ is EKR}\}.$$

Here an EKR graph is one whose maximum independent sets are precisely the stars, and a near-star is a family obtained from a star by deleting one member and adding one set outside the star. Hitting-time conjecture. With high probability, for all $n>2k+1$,

$$\tau_\alpha=\tau_{\textnormal{super}}\quad\text{and}\quad\tau_{\textnormal{near}}=\tau_{\textnormal{EKR}}.$$

The corresponding equalities are proved in the paper when $n=2k+1$; the conjecture asserts that the hitting-time phenomenon extends to all larger values of $n$ and $k$.