Hi Manu,
I think I may have figured this one out. Personally, with my models since I use some tiling textures I do a lot of UVs out of the 0,1 space. After checking those I could see the same issue in mine as well.
This appears to be a bug in the engine that after going back through previous engine versions it appears this has been this way and just not caught. I do want to point out that it isn’t destructive to your models UV even though it appears to be. I am going to put in a bug report for this. (TTP:347125)
Here is a screen shot of a model I created like yours that you’ve provided and looking the bottom piece that separates in the mesh viewer is fine as far as a texture is concerned.
I’ve applied a simple UV checker texture to see if this is being destructive but cannot see that it is.
Left is yours and right is mine
I hope this helps alleviate some of your frustrations in that it doesn’t appear to be Max/Maya’s FBX issue and it’s not destroying UVs as well.
If you have any concerns/thoughts or run into any other issues with this that you come along please let me know! 
Tim