Counterexample to a kernel-to-complete-intersection Macaulay dual generator claim

Determine whether there exists a counterexample to the claim that, for a squarefree monomial complete intersection ideal I ⊂ R not containing any pure power x_i^{a_i}, with integers a_i > 2 such that A = R/(I + (x_1^{a_1}, …, x_n^{a_n})) fails the WLP in degree t−1 and K = ker((×L)^T in degree t), there exists a polynomial F ∈ K that is the Macaulay dual generator of a complete intersection of the form I + (θ1, …, θr) where θ1, …, θr is a system of parameters for I.

Background

The authors propose a two-part question (Question 7.3) about when a linear combination of given polynomials can serve as a Macaulay dual generator of a complete intersection and, more specifically, whether such an F can always be found inside the kernel responsible for WLP failure for monomial complete intersections.

They remark that the second, more specific part likely overreaches but that they currently lack a counterexample, leaving open whether the strengthened statement is false.