The noncommutative Borsuk–Ulam conjecture for equivariant joins

Let $\mathbb{G}$ be a compact quantum group. The $n$-fold equivariant noncommutative join is denoted by $C(E^\Delta_n\mathbb{G})$, and $\dim_{\rm LT}^{\mathbb{G}}$ denotes the local-triviality dimension of a $\mathbb{G}$-algebra.

Noncommutative Borsuk–Ulam conjecture. There does not exist a $\mathbb{G}$-equivariant $*$-homomorphism

$$C(E^\Delta_n\mathbb{G})\to C(E^\Delta_{n+1}\mathbb{G})$$

and

$$\dim_{\rm LT}^{\mathbb{G}}(C(E^\Delta_n\mathbb{G}))=n.$$

This conjecture generalizes the classical Borsuk–Ulam non-existence result from finite and compact Hausdorff groups to compact quantum groups, in the setting of iterated equivariant noncommutative joins. The supplied text gives no resolution of the conjecture.