  Within the framework of free actions of compact quantum groups on unital
C*-algebras, we propose two conjectures. The first one states that, if $H$ is
the C*-algebra of a compact quantum group coacting freely on a unital
C*-algebra $A$, then there is no equivariant $*$-homomorphism from $A$ to the
join C*-algebra $A*H$. For $A$ being the C*-algebra of continuous functions on
a sphere with the antipodal coaction of the C*-algebra of funtions on
$\mathbb{Z}/2\mathbb{Z}$, we recover the celebrated Borsuk-Ulam theorem. The
second conjecture states that there is no equivariant $*$-homomorphism from $H$
to the join C*-algebra $A*H$. We show how to prove the conjecture in the
special case $A=C(SU_q(2))=H$, which is tantamount to showing the
non-trivializability of Pflaum's quantum instanton fibration built from
$SU_q(2)$.
