Hello, this morning I have discovered a new BB(2, 5) champion, dethroning the old champion
https://bbchallenge.org/1RB2LB4LB3LA1RZ_1LA3RA3LB0LB0RA with a sigma of >7.3 × 10^19016 steps!
But the gist is that the TM is part bouncer and part counter, but the counter can overflow, in which case the bouncer ends up being the counter. Since usually by that time, the size bouncer becomes an exponental of the size of the counter, having the bouncer turn into the counter essentially shoves whatever size the tape was to the exponent. This effect happens 3-4 times before the mod 3 value becomes the right condition to halt. At the end, both the sigma and step score can be bounded below by 2^^7 or 10^^4.