The QCA conjecture on the spectrum of quantum cellular automata

For each spatial dimension $d$, let $\mathcal Q^*(\mathbb Z^d)$ be the group of quantum cellular automata and let $\mathcal C^*(\mathbb Z^d)$ be its normal subgroup of quantum circuits. Let $\mathrm{QCA}(\mathbb Z^d)$ denote a proposed space-level construction of the QCA classification group. The QCA conjecture. There is a canonical isomorphism

$$\pi_0(\mathrm{QCA}(\mathbb Z^d))\cong \mathcal Q^*(\mathbb Z^d)/\mathcal C^*(\mathbb Z^d).$$

Moreover, these spaces assemble naturally into an $\Omega$-spectrum $\mathbb{QCA}=\{\mathrm{QCA}(\mathbb Z^d)\}_{d\ge 0}$. This conjecture would give a spectrum-level classification of QCA and organize their groups across spatial dimensions. The source describes substantial evidence and special cases, including Clifford QCA, but leaves the general statement open.