The noncommutative Brouwer retract conjecture for C*-algebras

Let $A$ be a unital $C^*$-algebra with a free action of $\mathbb{Z}/2\mathbb{Z}$, and let $\Gamma A$ be its cone C*-algebra. Let $\operatorname{ev}_{1/2}:\Gamma A\to A$ be evaluation at the midpoint.

Noncommutative Brouwer retract conjecture. There is no $*$-homomorphism

$$\gamma:A\longrightarrow\Gamma A$$

such that

$$\operatorname{ev}_{1/2}\circ\gamma=\operatorname{id}_A.$$

This is proposed as a noncommutative analogue of the retract formulation of Brouwer's fixed-point theorem and is an immediate consequence of the preceding cone Borsuk–Ulam conjecture. The source says it holds for the even-dimensional quantum $\theta$-ball families, but does not establish it in general.