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 of A and H.

When A is the C*-algebra of functions on a sphere, and H is the C*-algebra of functions on the group of two elements acting antipodally on the sphere,

then the conjecture of type I becomes the celebrated Borsuk-Ulam theorem.

I will recount some additional assumptions under which the conjectures hold true, few consequences of the type I conjecture, and a K-theoretic strengthening of the type II conjecture.