Universal B-representation with trivial C-ideal

Determine whether every C-representable ideal I admits a representation I = C(e) by a sequence e satisfying B(e) = P(N).

Background

The paper shows that B- and C-representations can be altered to use 1-unconditional bases, but the construction does not establish whether a C-representation can always be chosen whose weak unconditional convergence ideal is the entire power set.