Liu and Postle's density conjecture for triangle-free 4-critical graphs

A graph $G$ is 4-critical if it is $4$-colourable but every proper subgraph is $3$-colourable; it is triangle-free if it contains no subgraph isomorphic to $K_3$. Write $v(G)=|V(G)|$ and $e(G)=|E(G)|$. Liu and Postle's density conjecture. If $G$ is a $4$-critical triangle-free graph, then

$$e(G) \geq \frac{5v(G)+5}{3}.$$

This conjecture seeks a sharper lower-order improvement to the general Kostochka–Yancey edge bound for critical graphs after excluding triangles. The source gives no resolution status for the conjecture.