so, even though we still haven't completely clarified in our own mind what we're doing here, let's try looking at the invariant distributions on the a3 total flag variety as down-sets in a certain poset again, and let's try to identify the integrable ones, and enumerate the ones extending a given integrable one...

the down-sets:

[],[12],[23],[34],[12 23],[12 34],[23 34],[12 23 34],[13 12 23],[13 12 23 34],[24 23 34],[12 24 23 34],[13 12 23 24 34],[14 13 24 12 23 34]

the integrable ones:

[],[12],[23],[34],...???...