Independence number conjecture for sparse random Cayley graphs

Let $G$ be a group of size $n$, and let $G(p)$ be its random Cayley graph, formed by selecting each equivalence class $\{x,-x\}$ independently with probability $p$. Write $\alpha(G(p))$ for its independence number, and use $\tilde{O}$ to suppress polylogarithmic factors in $|G|$. Independence number conjecture. The independence number of the random Cayley graph $G(p)$ is at most $\tilde{O}(p^{-1})$ whp. This predicts that sparse random Cayley graphs have independence numbers comparable, up to polylogarithmic factors, to random regular graphs of the same degree. The statement is presented as a conjecture and no resolution is given in the source.