An unsolved problem: how many solutions are there to the 8-Color Cube?

The 8-Color Cube is an extremely elegant problem, both in appearance and concept;

The cube is very easy to make at home: numbered stickers are available everywhere and the whole
construction process takes only about 15 minutes.

As you can see, Walter Randelshofer and myself have managed to find a number of extra solutions
separate from the pre-existing design solution with its ”Superflip Centre” variant, however the real
problem remains: how many solutions are there, in theory, to the 8-Color Cube?--this is the tough
and so-far unanswered question that we would like to ask our honored fellow-members of the Cube
Forum to consider.

The problem seems easy: ”How hard can that be?!” one thinks...initially(!) actually turns out to
be pretty intractable...