Heisenberg's uncertainty principle is just an application of the Fourier uncertainty principle to the quantum position and momentum wavefunctions. Since those are a Fourier pair, the Fourier uncertainty principle applies. So it's not a matter of which interpretation you pick, it's about whether QM is correct. If the behavior of quanta is not wholly determined by their wavefunctions then QM is wrong, and Heisenberg's uncertainty principle might be violated, since the true functions for position and momentum may not be Fourier pairs. This is widely considered to be extremely unlikely. Also, if spacetime is quantized (there's a minimum possible distance and a minimum possible time, and all times/distances are integer multiples of these minima) then the wavefunctions wouldn't be continuous, so the uncertainty principles might not be applicable. Loop Quantum Gravity is one theory that posits this.