This is a special case of a more general problem, which is equally
open.
The goal, as in Problem 9, is to cut the surface
and unfold without overlap.
An edge unfolding only permits cutting along edges of the
polyhedron.
A grid unfolding
adds extra edges to the surface by intersecting
the polyhedron with planes parallel to coordinate planes
through every vertex, and so is easier to edge-unfold.
Easier still is the posed problem: The orthogonal polyhedron
is built from cubes, and all cube edges are available for cutting.
Is there any such polyhedron that cannot be edge-unfolded?
Such an example would narrow the options, but it may be that
every orthogonal polyhedron can be grid-unfolded.
(An easy box-on-box example [BDD+98] shows
that without some surface refinement [DO05],
not all orthogonal polyhedra
can be edge-unfolded.)
The posed question is among the most specific whose answer would
make progress.
Only a few narrow subclasses of orthogonal polyhedra
are known to have grid-unfolding algorithms:
orthotubes, orthostacks of orthogonally convex slabs,
and orthogonal terrains.
See [O'R08].