Complementary Betti numbers for monomial ideals

Establish whether every monomial ideal I with a nonzero multigraded Betti number β_{a+b,lcm(I)}(S/I) for positive integers a and b has complementary nonzero multigraded Betti numbers β_{a,m}(S/I) and β_{b,m'}(S/I).

Background

The question translates the lattice-complementation problem into the language of multigraded Betti numbers of monomial ideals. Complementarity requires lcm(m,m')=lcm(I), gcd(m,m') not in I, and nonvanishing Betti numbers in homological degrees a and b whose sum is the degree of the global Betti number.

The paper proves a substantial partial result: for every proper lattice element m, suitable complementary Betti numbers can be found after possibly passing to a divisor n of m. It also explicitly states that a full answer is not known and gives examples showing that not every individual Betti number must be complemented.