Broken Gorenstein characterization conjecture for smoothable points in the affine threefold Hilbert scheme

Let $S={\Bbbk}[x,y,z]$, let $d$ be a positive integer, and let $[S/I]\in\mathrm{Hilb}^d(\mathbb{A}^3)$ be a smoothable point. Broken Gorenstein characterization conjecture. The following conditions are equivalent: (1) the algebra $S/I$ admits a broken Gorenstein structure; (2) the point $[S/I]$ is smooth on the Hilbert scheme; and (3) the ideal $I$ is licci. This conjecture would characterize smoothable smooth points simultaneously by broken Gorenstein structures and licci ideals; the supplied text gives no resolution.