  All planar graphs are 4-colorable and 5-choosable, while some planar graphs
are not 4-choosable. Determining which properties guarantee that a planar graph
can be colored using lists of size four has received significant attention. In
terms of constraining the structure of the graph, for any $\ell \in
\{3,4,5,6,7\}$, a planar graph is 4-choosable if it is $\ell$-cycle-free. In
terms of constraining the list assignment, one refinement of $k$-choosability
is choosability with separation. A graph is $(k,s)$-choosable if the graph is
colorable from lists of size $k$ where adjacent vertices have at most $s$
common colors in their lists. Every planar graph is $(4,1)$-choosable, but
there exist planar graphs that are not $(4,3)$-choosable. It is an open
question whether planar graphs are always $(4,2)$-choosable. A chorded
$\ell$-cycle is an $\ell$-cycle with one additional edge. We demonstrate for
each $\ell \in \{5,6,7\}$ that a planar graph is $(4,2)$-choosable if it does
not contain chorded $\ell$-cycles.
