Planar graph with exclusively real and a non-integral chromatic root

Determine whether there exists a planar graph whose chromatic polynomial has only real roots and at least one non-integral chromatic root.

Background

The paper notes that chordal graphs have only nonnegative integer chromatic roots, while some non-chordal graphs have all integer chromatic roots and some non-planar graphs have exclusively real but non-integral chromatic roots. This motivates asking whether the analogous phenomenon can occur for planar graphs. The problem is explicitly identified as remaining widely open.