Type 1 noncommutative Borsuk–Ulam conjecture for compact group actions

Let $G$ be a nontrivial compact group acting freely on a unital $C^*$-algebra $A$. Define the join

$$A\circledast C(G)=\{f\in C([0,1],A\otimes C(G)): f(0)\in \mathbb{C}\otimes C(G),\ f(1)\in A\otimes\mathbb{C}\}.$$

Equip $A\circledast C(G)$ with the diagonal action of $G$. Type 1 noncommutative Borsuk–Ulam conjecture. There is no equivariant, unital $*$-homomorphism

$$\phi:A\to A\circledast C(G).$$

This is the compact-group subcase of the Type 1 noncommutative Borsuk–Ulam conjecture, originally formulated for coactions of compact quantum groups. It asserts that a free action cannot admit an equivariant unital map from $A$ to its join with the acting group.