We conducted ab initio molecular dynamics simulations of the d phase of the hydrous mineral aluminium oxide hydroxide (AlOOH) at ambient temperature and high pressure. Nuclear quantum effects were included through a Langevin dynamics in a bath of quantum harmonic oscillators. We confirm that under increasing pressure d-AlOOH undergoes a phase transition from a P2 1 nm structure with asymmetric and disordered O-H bonds to a stiffer Pnnm phase with symmetric hydrogen bonds, which should be stable within the pressure and temperature ranges typical for the Earth's mantle. The transition is initially triggered by proton tunneling, which makes the mean proton position to coincide with the midpoint of the O-O distance, at pressures as low as 10 GPa. However, only at much larger pressures, around 30 GPa as previously found by other calculations, the Pnnm phase with symmetric hydrogen bonds is stable from the classical point of view. The transition is also characterized through the analysis of the H-O stretching modes, which soften considerably and fade out around 10 GPa in the P2 1 nm structure, when thermal and nuclear quantum effects are taken into account in the simulations. At variance, the harmonic picture is not adequate to describe the highly anharmonic effective potential that is seen by the protons at the transition. Finally, we propose that the picture of a dynamical transition to the high-symmetry and proton-centered Pnnm phase, which is brought about by the onset of proton tunneling, could be confirmed by quasi-elastic neutron scattering and vibrational spectroscopy under pressure.