The basis conjecture states: *every MZV can be written as a sum
of rational multiples of MZVs of the same weight whose exponent
strings involve only 2's and 3's*. This involves relations
like

in weight 7. Here is an explicit list of such relations through weight 8. The conjecture was verified through weight 22 by J. Blümlein et. al. (see the paper), and proved by F. Brown and D. Zagier.