The exact-one failure conjecture for the weak Lefschetz property

Let $CI(d,d,d,d)$ denote the parameter space of complete intersections generated by four forms of degree $d$. Say that a member fails WLP by more than one or fails WLP by exactly one according to the deficiency in maximal rank described in the paper. Let $X\subset CI(d,d,d,d)$ be the locus of complete intersections that fail by more than one, and let $Y$ be the locus of those that fail by exactly one.

Exact-one failure conjecture. If there is a complete intersection $J\in CI(d,d,d,d)$ that fails WLP by more than one, then there exists a complete intersection $I\in CI(d,d,d,d)$ that fails WLP by exactly one; more precisely,

$$X\subset\overline{Y}.$$

This is proposed as the reduction step needed to prove the general height-four WLP conjecture: it would suffice to analyze failures by exactly one. The paper does not resolve this conjecture.