[BurningFrog]

You can trim it down by counting in quarter moves.

That is, count a 180° move as two 90° moves. Which it is, though we usually don't think of it that way.

Assuming a random move length distribution, that would only leave a (2/3)^20 fraction of them, which is about 147000 positions.

[kevindamm]

That page also shows their results for quarter turns, there are 3 (or fewer) with 26 quarter turns being the shortest they can be solved in. 36 (or fewer) needing 25 quarter-turns.

https://www.cube20.org/qtm/