Gap conjecture for the circular chromatic index

Let $k$ be an integer with $k geq 2$. The circular chromatic index of a graph $G$ is denoted by $\chi'_c(G)$.

Gap conjecture for the circular chromatic index. For every integer $k\geq 2$, there exists an $\epsilon>0$ such that no graph $G$ satisfies

$$k-\epsilon<\chi'_c(G)<k.$$

Equivalently, the interval $(k-\epsilon,k)$ is a gap in the possible circular chromatic indices of graphs. The conjecture is proposed based on the results of the paper, including the established gap below $4$ for the class considered there; the general assertion for every integer $k\geq2$ remains open in the supplied text.