Postle's density conjecture for clique-free critical graphs

Let $k\geq 4$ and let $G$ be a $k$-critical graph, meaning that $\chi(G)=k$ while every proper subgraph has chromatic number $k-1$. Write $e(G)=|E(G)|$ and $v(G)=|V(G)|$.

Postle's density conjecture. For every $k\geq 4$, there exists $\varepsilon_k>0$ such that if $G$ does not contain a $K_{k-2}$ subgraph, then

$$e(G)\geq\left(\frac{k}{2}-\frac{1}{k-1}+\varepsilon_k\right)v(G)-\frac{k(k-3)}{2(k-1)}.$$

This strengthens the Kostochka–Yancey lower bound for critical graphs under the absence of a large clique; the source presents it as a conjecture of Luke Postle, and no resolution is given here.