Simplify3D, connection disapear after slice
I've tried that item as well.  In fact, other than the heart, I built a remix  which works better but is still difficult to free up after printing.  Those tabs you're concerned about will print better at a very low (thin) layer setting. 

I am pretty sure that the original design is faulty.  All the "leaves" have gears which cause them to rotate when the threaded ring is turned.  But the designer made all leaves identical!  Each one needs its gear's  null-position set differently so that it properly meshes with the thread positions of the outer ring at that leaf's placement.  Some day I may remix to adjust those gear positions....