Flexibility of planar graphs of girth at least six

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at least a constant fraction of the preferences.
Comment: 11 pages, no figures. arXiv admin note: text overlap with arXiv:1902.02971