General upper-bound conjecture for interval cyclic colorings

Let $G$ be a connected graph with at least two vertices, let $V(G)$ be its vertex set, and let $\mathfrak{N}_{c}$ denote the class of interval cyclically colorable graphs. Write $W_c(G)$ for the maximum number of colors in an interval cyclic coloring. The general interval cyclic upper-bound conjecture.

$$W_c(G)\leq 2\lvert V(G)\rvert-3.$$

This conjecture proposes removing the maximum-degree term from the previously proved general bound for connected interval cyclically colorable graphs. The paper also relates it to the triangle-free case, but does not establish either conjecture.