A few years ago in a landmark paper, Guionnet and Shlyakhtenko proved the existence of free monotone transport maps from the free group factors to von Neumann algebras generated by elements which have a joint law ``close'' to that of the free semicircular law. In this talk, I will discuss how to modify their idea to obtain similar results for interpolated free group factors using an operator-valued framework. This is joint work with Brent Nelson.