I think it is the way that the surface was made.

The blue surface is your original and the green one is the offset of 1.5mm. Since you had mentioned thickening failed in a few packages already I went with offset to try to avoid a failure. Sure enough, the model ends up with open faces. I am not sure what your modeling process was but there is something about how the surfaces were trimmed during your process. This might be able to be solved by untrimming it but I will yield to any number of the more expert surface people on the blog.

I did try a few things such as seperating the surfaces and then trying to use the edges for closing off the surfaces but got strange results. Someone might know what is going on by looking at the control cage.