Actually, a fully convective gad giant is dynamic AND differentiated. Look at the colour variation on Jupiter - that's chemical or phase differentiation for a start. There are so many possible phase changes with pressure and temperature that differentation is inevitable in a convecting body.

Now, the convection may be so strong as to maintain mixing despite the differentiation but you would STILL have a layered body with different minerals/crystals/droplets/fluids at different vertical levels.

A body would only be undifferentiated if it was identical (or equally random) in chemical composition and crystal/droplet size at all levels of its core, mantle, crust, and atmosphere AND this setup was the way it assembled.