Does the STOP-cube have a second type of solution?

You might ask: "What the heck is a STOP-cube?" and "Why would I care? Who are you anyways?"
And those are all legit questions of course.

My name is Ortwin, I am a long time member of the twisty puzzle forum. Currently I am looking for the answer to question regarding a specific sticker modification of a 3x3x3 cube, but there did not seem to be much interest over there in the question. Walter Randelshofer who is also a member of this forum recommended to post it into this community, and hereby I do.

To get an idea what that "STOP-cube" is, you might want to have a quick look at the links to the topics in the twistypuzzle forum:

Start here. The question I am interested is here.

Let me try to summarize and rephrase what the Stop- cube is and what I am after:

The STOP-cube is a sticker variation that uses only one type of sticker on all 54 facelets of the cube. I chose the well known stop sign as design for that sticker because due to its octagonal shape it can "naturally" be applied in 8 different orientation on a given square facelet.
But really it could be an arrow instead or indeed any shape that does not have rotational symmetries.

In the solved state all the stickers on a respective side of the cube have the same orientation. The thing about it though is, that still more configurations are distinguishable than on a regular cube: half as many as on a supercube.

That is all good and well, but recently I began to suspect that there might be second type of solution to a cube with this type of stickers. That second type of solution should show all 8 possible orientations of the sticker on every side of the cube, the center facelets should not be stickered, it would correspond to a void cube.

There are some aspects of this problem that resemble the 8 color cube that was discussed (and the problem was solved I believe) here before.

It might be that for the second type of solution to be possible, the stickers need to be applied differently from my version and that is fine as long as the first type of solution would still be possible and the number of distinguishable positions does not become smaller.