The noncommutative Brouwer fixed-point conjecture for compact spaces

Let $X$ be a compact space of finite covering dimension equipped with a free action of $\mathbb{Z}/2\mathbb{Z}$. Let $\Gamma X$ denote the cone on $X$.

Noncommutative Brouwer fixed-point conjecture. Every continuous map

$$f:\Gamma X\longrightarrow\Gamma X$$

has a fixed point.

This proposes a Brouwer fixed-point theorem for cones over compact spaces with free involutions, without assuming an underlying linear structure. The source presents it as a conjectural extension of the classical Brouwer theorem; no resolution is supplied.