Acyclic $3$-choosability of planar graphs with no cycles of length from $4$ to $11$

Every planar graph is known to be acyclically $7$-choosable and is conjectured to be acyclically $5$-choosable (Borodin et al., 2002). This conjecture if proved would imply both Borodin's acyclic $5$-color theorem (1979) and Thomassen's $5$-choosability theorem (1994). However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically $4$- and $3$-choosable.
In particular, a planar graph of girth at least $7$ is acyclically $3$-colorable (Borodin, Kostochka and Woodall, 1999) and acyclically $3$-choosable (Borodin et al., 2010). A natural measure of sparseness, introduced by Erdős and Steinberg, is the absence of $k$-cycles, where $4\le k\le C$. Here, we prove that every planar graph with no cycles of length from $4$ to $11$ is acyclically $3$-choosable.