The density conjecture for 5/2-critical graphs

Let $G$ be a $5/2$-critical graph, meaning that $G$ has no circular $5/2$-coloring (equivalently, no homomorphism to $C_5$), while every proper subgraph of $G$ has such a coloring. Write $n(G)=|V(G)|$ and $e(G)=|E(G)|$.

The density conjecture for $5/2$-critical graphs. If $G$ is a $5/2$-critical graph, then

$$14n(G)-11e(G)\le 9.$$

The conjectured bound is tight for the graphs arising from the stated Ore construction and for the triangle. It is motivated by the asymptotic edge density $14/11$ of those examples, but the source does not state that the bound has been proved or disproved.