Feige–Pauzner conjecture for enabling graphs

Let $G$ be a graph on $n$ vertices, and let $k(n)$ be the maximum integer $k$ such that every vertex of $G$ is contained in both a clique of size $k$ and an independent set of size $k$.

Feige–Pauzner conjecture. For all $n>0$,

$$k(n)=\left\lfloor\frac{n}{4}\right\rfloor+1.$$

This conjecture gives the exact value of the largest common clique and independent-set size that can be locally guaranteed in an $n$-vertex graph. The paper states that it proves this conjecture, so its status is solved.