PSPACE-completeness conjecture for the on-line chromatic number

Let $G$ be a graph and let $k\in\mathbb{N}$. The on-line chromatic number conjecture. Deciding whether

$$\chi^O(G)\leq k$$

is \textsc{PSPACE}-complete.

The complexity of deciding whether the on-line chromatic number satisfies this bound was open in the paper; it was known to be \textsc{coNP}-hard and to lie in \textsc{PSPACE}. The paper proves \textsc{PSPACE}-completeness when a pre-coloring is supplied, motivating this conjecture for the unprecolored problem.