Conjecture on random Kneser graph chromatic number

Let $KG_{n,k}$ be the Kneser graph on the $k$-element subsets of $[n]$, and let $KG_{n,k}(p)$ be its random subgraph obtained by retaining each edge independently with probability $p$. Write $\chi$ for chromatic number. Random Kneser graph conjecture. For every fixed $p>0$,

$$\chi\bigl(KG_{n,k}(p)\bigr)\ge \chi\bigl(KG_{n,k}\bigr)-4$$

whenever $k\gg\log n$. This conjecture proposes that, in the indicated range, passing to a random subgraph lowers the chromatic number by at most an additive constant; the supplied passage states that the corresponding question is open and gives no resolution of this conjecture.