The improper 4-coloring conjecture for toroidal graphs with defect sequence $(0,0,0,3)$

A toroidal graph is a graph that can be embedded on the torus. A coloring is $(0,0,0,3)$-improper when the vertices are partitioned into four color classes such that the first three induce graphs of maximum degree $0$, while the fourth induces a graph of maximum degree at most $3$.

Improper coloring conjecture for toroidal graphs. Every toroidal graph is $(0,0,0,3)$-colorable.

The paper proves this property for every not $5$-degenerate toroidal graph, so any counterexample would have to be $5$-degenerate. The conjecture is presented as open; it would sharpen the known improper-coloring results for toroidal graphs.