Inspired by the 3D printer weaving technique by /u/CheesmondN, I created a Weaving Test myself and figured out that the filament doesn't need to be weaved to stick to the base model. By wrapping the filament around the base model, a unique laced effect can be created without supports!
I've provided 4 profiles with the settings required to achieve the laced effect:
The Multi-Color profile has a Flushing Volume Multiplier of 0. The eggs printed in the model image were printed without flushing to minimize waste, save time, and blend the colors a bit when switching. The profiles also must have “Detect Thin Walls”.
Lastly, the FreeCAD file is provided for those interested in how I modelled it.