Circular-colouring density conjecture for 4-critical graphs

Let $p$ and $q$ be integers satisfying $3<\frac{p}{q}<4$. A $(p,q)$-colouring is a graph homomorphism to the circular clique with vertex set $\{0,1,\ldots,p-1\}$, in which two vertices $i,j$ are adjacent when $q\leq|i-j|\leq p-q$. Let $G$ be a $4$-critical graph, meaning that $\chi(G)=4$ while every proper subgraph has chromatic number $3$.

Circular-colouring density conjecture. If $G$ has no $(p,q)$-colouring, then there exist positive rational numbers $\varepsilon_{p,q}$ and $c_{p,q}$, depending on $p$ and $q$, such that

$$e(G)\geq\frac{(5+\varepsilon_{p,q})-c_{p,q}}{3}.$$

The source describes this as a strong conjecture motivated by structural results on $4$-critical graphs without such colourings. The displayed bound appears exactly as stated in the source, and no resolution is given.