Dvořák–Postle edge conjecture for $C_5$-critical graphs

For a graph $H$, an $H$-critical graph is a graph that admits no homomorphism to $H$, while every proper subgraph does. Here $C_5$ denotes the cycle on five vertices, and $e(G)=|E(G)|$, $v(G)=|V(G)|$.

Dvořák–Postle conjecture. If $G$ is $C_5$-critical, then

$$e(G)\geq\frac{14v(G)-9}{11}.$$

This conjecture concerns the density of graphs critical for homomorphisms to the five-cycle and is stated after a weaker proven bound in the source; no resolution is supplied here.