Asymptotic online coloring of the graph $C_5^+$

Let $C_5^+$ be the graph obtained from the 5-cycle $C_5$ by adding the edge indicated in the paper, and let $f(w,G)$ denote the maximum number of colors that an adversary can force a deterministic online coloring algorithm to use when the clique number of the token graph is at most $w$. Asymptotic coloring conjecture.

$$f(w,C_5^+)=\left(\frac{5}{4}+o(1)\right)w.$$

This is presented as an approachable open problem concerning the asymptotic behavior of the online coloring parameter for the remaining minimally non-online-perfect graphs; the paper does not establish the claimed asymptotic.