The tournament scheduling conjecture for on-line scheduling

Let $G$ be a non-edgeless multigraph, let $t\in\mathbb{N}$, and let $\chi_{\textit{OL}}^t(G)$ denote the worst-case number of rounds required for on-line scheduling when each vertex is allowed $t$ absences. Let $\chi'(G)$ be the chromatic index of $G$. On-line scheduling conjecture. For every such $G$ and $t$,

$$\chi_{\textit{OL}}^t(G)=\chi'(G)+2t.$$

This is described as a weakening of the on-line List Edge Coloring Conjecture and would determine the on-line scheduling number for all non-edgeless multigraphs once the chromatic index is known. The source gives no resolution; it notes that the equality is known in several special cases.