  Let $H$ be the C*-algebra of a non-trivial compact quantum group acting
freely on a unital C*-algebra $A$. It was recently conjectured that there does
not exist an equivariant $*$-homomorphism from $A$ (type-I case) or $H$
(type-II case) to the equivariant noncommutative join C*-algebra
$A\circledast^\delta H$. When $A$ is the C*-algebra of functions on a sphere,
and $H$ is the C*-algebra of functions on ${\mathbb Z}/2{\mathbb Z}$ acting
antipodally on the sphere, then the conjecture of type I becomes the celebrated
Borsuk-Ulam theorem. Following recent work of Passer, we prove the conjecture
of type I for compact quantum groups admitting a non-trivial torsion character.
Next, we prove that, if a compact quantum group admits a representation whose
\mbox{$K_1$-class} is non-trivial and $A$ admits a character, then a stronger
version of the type-II conjecture holds: the finitely generated projective
module associated with $A\circledast^\delta H$ via this representation is not
stably free. In particular, we apply this result to the $q$-deformations of
compact connected semisimple Lie groups and to the reduced group C*-algebras of
free groups on $n>1$ generators.
