Dąbrowski's suspension conjecture for free involutions

Let $A$ be a unital $C^*$-algebra with a free action of $\mathbb{Z}/2\mathbb{Z}$. Its unreduced suspension is

$$\Sigma A=\{f\in C([-1,1],A):f(-1),f(1)\in\mathbb{C}\}.$$

The action extends to $\Sigma A$ by $\alpha_\Sigma f(t)=\alpha(f(-t))$. Dąbrowski's conjecture. There is no $\mathbb{Z}/2\mathbb{Z}$-equivariant $*$-homomorphism

$$\phi:A\to\Sigma A.$$

This conjecture seeks a noncommutative analogue of the Borsuk–Ulam obstruction and was proposed in connection with noncommutative Brouwer fixed point theorems. The source gives no resolution status.