WLP for (a, b)-geproci sets

Determine for which integers d and which (a, b)-geproci sets X ⊂ P^3 the Artinian algebra R/A_{X,d}, where A_{X,d} is the ideal generated by the d-th powers of the linear forms dual to the points of X, has the Weak Lefschetz Property.

Background

A set X ⊂ P^3 is (a, b)-geproci if its general projection to P^2 is a complete intersection of type (a, b). Grids are special examples of geproci sets that lie on a smooth quadric, enabling precise control of projections and the Hilbert functions of powers of complete intersection ideals.

The authors provide general cokernel-dimension formulas for multiplications by a general linear form in the geproci setting (Lemma 4.1), but beyond grids the structure is more delicate because most geproci sets do not lie on a quadric. This question seeks a characterization of when R/A_{X,d} has WLP across the broader geproci class.