Sounds like the colors ARE being exported to the STEP file, but SW is NOT importing them.
One story I heard years ago when IGES was being worked on and a vendor (remaining unnamed to protect the guilty) was doing their import demonstration. They brought in a surface and each segment of the UV grid was an individual line, so a 3x3 UV ended up with 18 line segments instead of 6. When asked why they did it that way the answer was "No one said the surface grid had to be useable".