Is your VMT in the folder materials\models\weapons\c_models\slicker? Does it match the name of the Blender material?

Also, the VTF path looks very wrong. It should be something like "materials/models/weapons/the_slicker-phys". I don't think the engine will load materials unless they're in the materials directory.

LightmappedGeneric is the shader for static BSP geometry. You want VertexLitGeneric.

Post the contents of your VMT file.

You'll need to put the VMT and VTF in the same folder as the existing Alyx materials. Make sure that the filenames match the material name used in Blender.

Only material/texture names are exported. It's up to you to create the VMTs and VTFs and put them in the right folder.

I can import it just fine. Does the console (Window > Toggle System Console) say anything?

Please upload the SMD somewhere.

Have you tried using the Trianguate modifier? There might be a setting in there that fixes this problem.

There are two fixes that I can think of.

The first is to triangulate before splitting shapes into separate objects, but that's a bad idea as it can lead to drastic changes to the output (e.g. per-face dupligroups).

The other is to recombine all of the shape objects to one mesh after triangulation, then split them off into separate objects again. That should work, but would make exports considerably slower. Given that this is an edge case in a legacy export format, I'm not happy about that prospect. I may experiment with it and see how bad the slowdown is though...