Question: Simplify equation under a parameter constraint

Dear Maple community,

I'm trying to simplify the attached Expression "under several parameter constraints", see simplify_under_constraint.mw

The constraints are threefold:

1. pi[1, 1, 1, F] + pi[2, 1, 1, F] + pi[3, 1, 1, F] = 1,

2. pi[1, 2, 1, F] + pi[2, 2, 1, F] + pi[3, 2, 1, F] = 1

3. pi[1, 3, 1, F] + pi[2, 3, 1, F] + pi[3, 3, 1, F] = 1,

It seems like, whenever the constraints appear in the exact order as specified above, simplify(Expression, {(pi[1, 3, 1, F] + pi[2, 3, 1, F] + pi[3, 3, 1, F]) = 1, (pi[1, 1, 1, F] + pi[2, 1, 1, F] + pi[3, 1, 1, F]) = 1, (pi[1, 2, 1, F] + pi[2, 2, 1, F] + pi[3, 2, 1, F]) = 1}) can somewhat simplify the Expression. However, is there a way for Maple to recognize the above-mentioned constraints for all possible "permutations" of, say, pi[1, 3, 1, F] = 1 - pi[2, 3, 1, F] - pi[3, 3, 1, F]; pi[2, 3, 1, F] = 1 - pi[1, 3, 1, F] - pi[3, 3, 1, F], etc. and simplify the expression in the most compact manner.

Thank you.