List $2$-arboricity of planar graphs with no triangles at distance less than two

It is known that not all planar graphs are $4$-choosable; neither all of them are vertex $2$-arborable. However, planar graphs with no triangles at distance less than two are known to be $4$-choosable (Lam, Shiu, Liu, 2001) and $2$-arborable (Raspaud, Wang, 2008).
We give a common extension of these two last results in terms of covering the vertices of a graph by induced subgraphs of variable degeneracy. In particular, we prove that every planar graph with no triangles at distance less than two is list $2$-arborable.