Yes, the cubex has issues with printing layers over supports.
My prints turn out stringy like that too when it has to print over supports.
There are a couple of things you can do:
First: Try rotating your print and turning off supports.
Put the wide curvy parts on the bottom and let it build the thin part vertically.
I think it can handle the angle of the curves without supports.
You may need a raft for it to build on top of, but try it without the raft first.
Printing without supports is preferred - the Cubex does a great job on builds that do not require supports.
If you have to have supports:
Print all one color - supports and build. If the print heads are not perfectly aligned, it can miss support edges and print in air
making a very messy stringy print.
Use support lines instead of points. The newer versions of the firmware (after 1.05) handle supports better.
Older versions would fuse the supports into the build, making it impossible to remove.
You can design your own supports.
Put them closer together than Cubex does, that will help with drooping sections and with break-away issues.