NP-intermediate F-free colouring problems

Determine whether, assuming P ≠ NP, there exists a finite set F of 2-edge-coloured graphs for which the F-free 2-edge-colouring problem is NP-intermediate.

Background

The authors explain that arbitrary forbidden 2-edge-coloured graphs lead to an extension of generalized monotone SNP that permits inequality, and that this broader logic can capture NP-intermediate problems unless P = NP. They therefore ask whether an F-free colouring problem can realize such intermediate complexity rather than falling into the usual polynomial-time or NP-complete categories.