The surface \ce 3-system list-colouring conjecture

Let $S$ be a fixed surface. For a graph $G$ embeddable on $S$ with a $\Sigma$-system, define

$$k(G;\Sigma)=\max_{u\ne v}|\Sigma(u)\cap\Sigma(v)|$$

and let $\Delta(G;\Sigma)=\max_{v}|\Sigma(v)|$. The surface $\Sigma$-system list-colouring conjecture. There exists a constant $c_S$ such that, for every graph $G$ embeddable on $S$,

$$\mathit{ch}(G;\Sigma)\le\Delta(G;\Sigma)+k(G;\Sigma)+c_S.$$

This aims to combine the paper's main theorem with a degeneracy-based list-colouring bound, extending the latter to surface-embeddable graphs. The source gives no resolution.