Caviglia–VandeBogert–Weyman conjecture on licci ideals and Dynkin formats

Let $R$ be a Gorenstein local ring and let $\mathfrak{f}=(1,m,m+n-1,n)$ be a resolution format realized by some grade $3$ perfect ideal in $R$. A perfect ideal is licci if it belongs to the linkage class of a complete intersection. Caviglia–VandeBogert–Weyman conjecture. If $\mathfrak{f}$ is not a Dynkin format, then there exists a grade $3$ perfect ideal in $R$ of format $\mathfrak{f}$ that is not licci; if $\mathfrak{f}$ is a Dynkin format, then every grade $3$ perfect ideal in $R$ of format $\mathfrak{f}$ is licci. Dynkin formats are the formats associated with the finite-dimensional Dynkin Lie algebras listed in the surrounding discussion. The conjecture describes the expected precise relation between Dynkin-type resolutions and linkage to complete intersections; the supplied source gives no resolution status.