Pavel found an Ackermann level BB(7) TM last weekend. I evaluate that it halts with a sigma score between
2{11}2{11}3 and
2{11}2{11}4 (where
{k} represents k Knuth up-arrows). See my analysis at
https://wiki.bbchallenge.org/wiki/1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF
Note: Please take it with a grain of salt, I have not double checked all the steps and there may be some errors (users on the Discord already noticed one off-by-one error).
Pavel reports that they found this machine after enumerating only a small portion of the BB(7) domain, so presumably there may be many more Ackermann level machines like this.