Baum-Dąbrowski-Hajac noncommutative Borsuk-Ulam conjectures

Let $A$ be a unital $C^*$-algebra with a free coaction $\delta$ of a nontrivial compact quantum group $(H, \Delta)$. Equip the join $A \circledast^{\delta} H$ with the induced free coaction $\delta_\Delta$. Baum-Dąbrowski-Hajac's noncommutative Borsuk-Ulam conjectures. There are no equivariant unital $*$-homomorphisms

$$A \to A \circledast^{\delta} H$$

that are $(\delta, \delta_\Delta)$-equivariant, and

$$H \to A \circledast^{\delta} H$$

that are $(\Delta, \delta_\Delta)$-equivariant. Equivalently, both the Type 1 and Type 2 conjectures in the source assert the nonexistence of the respective homomorphisms. These conjectures extend the topological formulation to coactions of compact quantum groups on possibly noncommutative $C^*$-algebras. The source gives no resolution status for either type.