WLP for R/A_{X,d} when X is a complete intersection

Ascertain general results about the Weak Lefschetz Property for the Artinian algebra R/A_{X,d} when X ⊂ P^3 is a zero-dimensional complete intersection and A_{X,d} is generated by the d-th powers of linear forms dual to the points of X.

Background

In the grid setting, the strength comes from the fact that a general projection is a complete intersection, giving accessible Hilbert functions for powers. In contrast, when X itself is a complete intersection in P^3, general projections are not controlled, but the Hilbert functions of symbolic powers of I_X are easy to compute.

This question inverts the paper’s usual approach: with symbolic powers well-understood for complete intersections, can one leverage that to deduce WLP behavior for R/A_{X,d}?