SketchUp-generated DAE files with multiple textures missing faces after import

Hello all.

I'm trying to develop a free workflow to work on a small project, just to have something in a portfolio when I graduate. I've been blundering along with Google SketchUp to do some level building, and the tools are alright for making the areas. I've been testing this by making a few simple two-room buildings and applying Google's built-in textures.

However, when I export these meshes to .dae files and subsequently import them into Unity, I am witnessing a very odd occurrence. Any textures beyond the first one applied to the model do not import (that is to say, Unity only generates one blank Material for a mesh that should have two), and more importantly, the faces with the missing textures are missing as well.

I have confirmed that the faces in question are indeed being saved from SketchUp by viewing the files with Quick Look on OS X (apparently you can even do a rudimentary fly-around with it).

Any ideas?