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.