Conjecture on licci ideals and nonzero structure maps

Let $I$ be a perfect ideal of height $3$ in a Gorenstein local ring $R$ with infinite residue field. For a structure map $w^{(i)}_{j,k}(J)$, write “nonzero modulo the maximal ideal” for nonvanishing after reduction modulo the maximal ideal of $R$. Structure-map criterion conjecture. The following are equivalent: (1) $I$ is licci; (2) for every ideal $J$ in the linkage class of $I$, there exists a structure map $w^{(i)}_{j,k}(J)$, with $i=1,2,3$ and $j\geq1$, that is nonzero modulo the maximal ideal of $R$; (3) there exists a structure map $w^{(1)}_{j,k}(I)$, with $j\geq1$, that is nonzero modulo the maximal ideal of $R$. Here licci means linked to a complete intersection by finitely many links. This conjecture proposes a characterization of licci ideals through higher structure maps and extends the preceding sufficient conditions; the supplied source gives no resolution status.