Boundedness of WLP exponent vectors

Determine whether, for a fixed d-dimensional Cohen–Macaulay orientable pseudomanifold over a field K and sufficiently large dimension d, the set of exponent vectors for which the monomial artinian reduction has the weak Lefschetz property is bounded, and determine whether the condition d at least 4 suffices for boundedness.

Background

For a d-dimensional Cohen–Macaulay orientable pseudomanifold Delta, the paper considers monomial artinian reductions obtained by adjoining arbitrary powers of the variables to the Stanley–Reisner ideal: I_Delta(a_1,...,a_n) = I_Delta + (x_1^{a_1},...,x_n^{a_n}). The set X_Delta records precisely those exponent vectors for which the resulting quotient has the weak Lefschetz property.

The paper proves WLP failure for an important family of exponent vectors, namely the uniform choice a_i=d+2, and notes that monomial almost complete intersections and related ideals can exhibit varied Lefschetz behavior. The question asks whether, in high dimensions, WLP can occur only for a bounded range of variable exponents.