Accessibility of standard configurations in the countable edgeless cube

Determine whether every standard configuration of the countable edgeless cube is accessible from the solved configuration; if so, characterize exactly which configurations are accessible.

Background

For the countable edgeless cube, the paper establishes that every accessible configuration is standard and that standard configurations invariant under all face twists are accessible from every standard configuration in at most $\omega^2$ moves. It leaves unresolved whether the inclusion of the accessible configurations in the set of standard configurations is proper, equivalently whether inaccessible standard configurations exist.