The classification conjecture for three-dimensional translation-invariant Clifford QCAs

Let $p$ be a prime, and let $\mathfrak C(D,p)$ denote the group of translation-invariant Clifford quantum cellular automata in spatial dimension $D$, modulo Clifford circuits and shifts. Three-dimensional Clifford-QCA classification conjecture.

$$\mathfrak C(D=3,p=2)=\mathbb{Z}_2,$$ $$\mathfrak C(D=3,p\equiv 1\pmod 4)=\mathbb{Z}_2\times\mathbb{Z}_2,$$

and

$$\mathfrak C(D=3,p\equiv 3\pmod 4)=\mathbb{Z}_4.$$

The paper has established the indicated examples and exponent bounds, but states that the full determination of $\mathfrak C(D\geq 3,p)$ remains open; thus these equalities are presented as a conjectural classification.