The edge-density conjecture for $K_{k-2}$-free $k$-critical graphs

Let $k\ge 4$, let $G$ be a graph, and write $V(G)$ and $E(G)$ for its vertex and edge sets. The graph $G$ is $k$-critical if its chromatic number is $k$ and every proper subgraph has chromatic number less than $k$; it is $K_{k-2}$-free if it contains no clique on $k-2$ vertices.

Edge-density conjecture. For every $k\ge 4$, there exists $\epsilon_k>0$ such that if $G$ is a $k$-critical $K_{k-2}$-free graph, then

$$|E(G)|\ge \left(\frac{k}{2}-\frac{1}{k-1} + \epsilon_k\right)|V(G)| - \frac{k(k-3)}{2(k-1)}.$$

The conjecture asks whether excluding $K_{k-2}$ improves the tight general lower bound of Kostochka and Yancey for the number of edges in a $k$-critical graph; excluding $K_{k-1}$ is known not to improve that bound. The surrounding discussion identifies this as an open direction, while related asymptotic results are known for fixed-size excluded cliques.