Thomassen's finiteness conjecture for 5-list-critical graphs on surfaces

Let $d4a2$ be a surface, and let $G$ be a graph embedded in $d4a2$. A list assignment $L$ assigns a set of colors to each vertex; $G$ is $L$-critical if it has no $L$-coloring but every proper subgraph of $G$ has an $L$-coloring. Thomassen's conjecture. For every surface $d4a2$ there are, up to isomorphism, only finitely many graphs $G$ such that $G$ embeds in $d4a2$ and is $L$-critical for some $5$-list assignment $L$. This asks for a finite obstruction set for $5$-list-colorability on each fixed surface; the supplied text gives no resolution status.