PCT is the only model I am aware of which even attempts to bridge the gap in between.

FWIW, Hawkins's HTM model (described in "On Intelligence") makes another fair stab at it, and has many similar characteristics to some of PCT's mid-to-high layers, just from a slightly different perspective. HTM (or at least the "memory-prediction framework" aspect of it) also makes much more specific predictions about what we should expect to find at the neuroanatomy level for those layers.

OTOH, PCT makes more predictions about what we should see in large-scale human behavioral phenomena, and those predictions match my experience quite well.