  We equip the multi-pullback $C^*$-algebra $C(S^5_H)$ of a
noncommutative-deformation of the 5-sphere with a free $U(1)$-action, and show
that its fixed-point subalgebra is isomorphic with the $C^*$-algebra of the
multi-pullback quantum complex projective plane. Our main result is the stable
non-triviality of the dual tautological line bundle associated to the action.
We prove it by combining Chern-Galois theory with the Milnor connecting
homomorphism in $K$-theory. Using the Mayer-Vietoris six-term exact sequences
and the functoriality of the K\"unneth formula, we also compute the $K$-groups
of $C(S^5_H)$.
