Online Ohba conjecture

Let $G$ be a graph, with vertex number $|V(G)|$, chromatic number $c7(G)$, and on-line choice number ${\rm ch}^{\rm OL}(G)$. A graph is on-line chromatic-choosable when $\chi(G)={\rm ch}^{\rm OL}(G)$. Online Ohba conjecture. If

$$|V(G)|\leqslant 2\chi(G),$$

then

$$\chi(G)={\rm ch}^{\rm OL}(G).$$

This is an on-line analogue of Ohba's conjecture, which was known to hold with the bound $|V(G)|\leqslant 2\chi(G)+1$ but fails in the on-line setting at that bound. The conjecture was proposed by Harutyunyan, Wigderson and Zhu and remains unresolved here.