Bonamy–Kardos–Kelly–Postle conjecture on fractional arboricity of planar graphs

Let $G$ be a planar graph, and let $a_f(G)$ denote its fractional arboricity: the minimum ratio $p/q$ for which vertices can be assigned at least $q$ colors from a total of at most $p$ colors, with every color class inducing an acyclic subgraph. Bonamy–Kardos–Kelly–Postle conjecture. Every planar graph satisfies

$$a_f(G)\leq 2.$$

The paper's abstract states that this conjecture is refuted by a planar signed simple graph whose fractional balanced chromatic number is larger than 2; the construction also yields planar graphs with fractional arboricity exceeding 2.