C-representations with universally bounded B-ideal

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

Background

The paper shows that B- and C-representations can be normalized using unconditional bases, but the resulting B-ideal need not equal P(N). It remains unknown whether a C-representable ideal can always be represented with all subsets belonging to the corresponding B-ideal.