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

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

Background

Flow polynomials are presented as the natural counterpart of chromatic polynomials. By planar duality, the question about planar chromatic roots is equivalent to a corresponding question about planar flow roots. The paper explicitly states that this problem remains open, although the main theorem later provides a negative answer.