The clique-number conjecture for graphs with chromatic number near the order

Let $Q(n,r)$ denote the minimum clique number among graphs on $n$ vertices with chromatic number $r$. The clique-number conjecture. Let $k$ be a positive integer. If $n$ is sufficiently large, then

$$Q(n,n-k)=n-2k+\frac{k}{2}\text{ rounded up}.$$

This conjecture was disproven, although the stated formula was later verified for $k\leq 6$.