= Solution
The walls occur in three families of parallel lines, and a choice of halfspaces in one family is determined by an integer cut. Lines from different families cross, so the three cuts can be chosen independently. It follows directly from the <dual cube complex of a wallspace> construction that
$$
C^{(0)}\cong\mathbb Z^3
$$
and that $C$ is the standard cubulation of $\mathbb R^3$.
Choose affine coordinates $x_1,x_2,x_3$ for the three wall families so that the original Euclidean plane is $x_1+x_2+x_3=0$ and the walls are the integer level sets. On the cut coordinates $(n_1,n_2,n_3)\in\mathbb Z^3$, the translation subgroup of the <Affine Coxeter group> $W$ adds vectors $(m_1,m_2,m_3)$ satisfying $m_1+m_2+m_3=0$, while its finite reflection subgroup permutes the three coordinates. Therefore
$$
h(n_1,n_2,n_3)=n_1+n_2+n_3
$$
is constant on every $W$-orbit. Since $h(n,0,0)=n$ is unbounded, there are infinitely many vertex orbits. A cocompact cubical action on this locally finite cube complex would have only finitely many cube orbits, so the action of $W$ on $C$ is not cocompact.
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