Continuity in q of quantum projective spaces for quantum Gromov–Hausdorff distance

Determine whether the compact quantum metric space structures on quantum projective spaces C(CP_q^r), arising from the D’Andrea–Dąbrowski spectral triples, depend continuously on the deformation parameter q in (0,1] with respect to Rieffel’s quantum Gromov–Hausdorff distance.

Background

The paper recalls that for the Podleś sphere (the r=1 quantum projective space) continuity in q has been established for Rieffel’s quantum Gromov–Hausdorff distance, but emphasizes that the analogous problem for higher-dimensional quantum projective spaces is unsettled.

This question concerns metric convergence of compact quantum metric spaces associated to D’Andrea–Dąbrowski spectral triples on quantum projective spaces under variation of the deformation parameter q.