I think it's hard to be favored to get it, although solos might be easier in this format than average because people have limited time and probably will prioritize the games with their own chances of doing well.

But, we could do a rough calculation. Suppose that we get 7 of each power and are equally skilled with every power. In that case, a player with a 28.7% win rate would have a 50% chance of soloing once with every power.

In practice, players aren't equally skilled with every power. So, a good distribution (for me, that would be a bunch of Italy, as I tend to draw a lot of my games with a big Italy, but fewer Russia, as I get eliminated a fair amount in gunboat but win a good fraction of the games when I get really big) would make an enormous difference. The problem is that you're very limited by your weakest combination of frequency and power. So, for example, a player who solos 28.7% of the time probably is not much better than 35% with their best country, and if you only get that country twice, you're already a slight underdog to solo one of those two times.

And on the third hand, if soloing once as each power is an explicit goal, it might mean that a player spends more of their focus on the games as harder powers, which in this sort of format might be very helpful in the midgame?