Tough question!
My guess:
I would expect the Vinca to win, if, once established, they manage to retain their foliage over winter. That's because the Viola, although possibly long-lived, do die eventually -- they tend to maintain their colonies by a progression of seedlings. So if the Vinca is dense enough all year, the Viola seedlings would have a job getting going. But if the Vinca does let light in over winter, the two could co-exist for a long time.

I've had vinca minor win against pansies, violas, strawberries, dianthus, and possibly some odorata violets (planning to move them!) They're trying unsuccessfully to overwhelm lavender and garlic chives, and it's a losing battle trying to keep them out of the mulch around the roses. Do you weed them out around the roses, or just let them fill in?

I let the vinca minor fill in around the roses. I don't want to disturb the roots of the roses. I will pile up mulch around the bases of the roes to protect them in the winter.

I think vinca minor and major would win against anything! It serves a wonderful purpose in specialized areas, but I would never, never place it in a flower bed. I have it around the house (which is on a raised mound of soil) next to the foundation and I can't get a shovel in it to plant a shrub. One year, I spot sprayed for dandelions in it and the next season it was thicker than ever. They only way I keep it from taking over the yard is mowing the edges regularly. The vinca major is now growing up through some globe arborvitae and hostas. I need to do something there. Be very careful where you place these plants.......

Update: years later vinca minor triumphant. Violas will not grow up through VM foliage. However Hellebores, daffodills, autmn crocus, roses, witchhazel, Rose of sharon, and daylillies can grou through VM.

As noted VM is a great spreader and tougher than creeping charlie. I would not put in into a flower border garden, but as an undershrub groundcover it is great. The only weed stronger than VM is sheep sorrel, but only in a region of poor soil.

Aha!
Thnx for the update Isabella.
I grow a double-flowered Vinca minor (curiosity more than anything) and a variegated one (much less drab under evergreen shrubs; Arbutus unedo in my case). Tough they are. If only all my violets were as tough...
...But then I guess I might find reasons to curse them!
...And that wouldn't do!

Just want to be complete!

I have had to replant violas on a cleared berm area, so that I can continue to enjoy them. I planted the violas with cape plumbago (leadwort). My ambition is that these two flowers can co-exist together and complement each others prime growth and flowering periods.