  The $K_0$-group of the C*-algebra of multipullback quantum complex projective
plane is known to be $\mathbb{Z}^3$, with one generator given by the C*-algebra
itself, one given by the section module of the noncommutative (dual)
tautological line bundle, and one given by the Milnor module associated to a
generator of the $K_1$-group of the C*-algebra of Calow-Matthes quantum
3-sphere. Herein we prove that these Milnor modules are isomorphic either to
the section module of a noncommutative vector bundle associated to the
$SU_q(2)$-prolongation of the Heegaard quantum 5-sphere $S^5_H$ viewed as a
$U(1)$-quantum principal bundle, or to a complement of this module in the
rank-four free module. Finally, we demonstrate that one of the above Milnor
modules always splits into the direct sum of the rank-one free module and a
rank-one non-free projective module that is \emph{not} associated with $S^5_H$.
