Graph with exclusively real and a non-integral flow root

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

Background

This is a weaker version of the planar flow-root problem, obtained by dropping planarity. It asks whether real-rootedness of a flow polynomial can occur without all flow roots being integers. The paper explicitly describes this weaker question as still unsolved; the main theorem subsequently resolves it negatively.