The even-order conjecture for self-homotopy equivalences of $A_2^2$-polyhedra

Let $X$ be an $A_2^2$-polyhedron, and let ${\mathcal B}^4(X)$ denote the group under consideration of self-homotopy equivalences of $X$. Even-order conjecture. If ${\mathcal B}^4(X)$ is a non-trivial finite group, then it necessarily has an element of even order.

The preceding theorem establishes strong necessary conditions for a finite group ${\mathcal B}^4(X)$ of odd order, but the authors report that they were unable to find any space satisfying those conditions. The conjecture asserts that such a non-trivial finite odd-order group is never realised in this setting.