With 25,000 QTM cosets proved to have a distance of 25 or less,
we have shown that there are no positions that require 30 or more
quarter turns to solve. All these sets were run on my personal
machines, mostly on a new single i7 920 box.
These sets cover more than 4e16 of the total 4e19 cube positions,
when inverses and symmetries are taken into account, and no new
distance-26 position was found. This indicates that distance-26
positions are extremely rare; I conjecture the known one is the
only distance-26 position.
In order to take the step to a proof of 28, I would need a couple
of CPU years, or improvements in my program or technique or both.
I will continue solving cosets and look for additional opportunities.
I believe a proof of 20 HTM and 26 QTM (or a counterexample!) will
probably happen within the next few years.