Petrosyan–Mkhitaryan conjecture on cyclic interval colorings

Let $\mathfrak{N}_c$ be the class of multigraphs admitting a cyclic interval coloring, and let $W_c(G)$ denote the maximum number of colors in a cyclic interval coloring of $G$. Petrosyan–Mkhitaryan conjecture.

(i) For every triangle-free graph $G\in\mathfrak{N}_c$,

$$W_c(G)\leq \lvert V(G)\rvert.$$

(ii) For every graph $G\in\mathfrak{N}_c$ with at least two vertices,

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

The paper states that the first bound is known for graphs of maximum degree at most $4$, while the general assertions are presented as conjectures; the second bound extends the corresponding general upper-bound question beyond triangle-free graphs.