Nonzero higher structure map conjecture for Dynkin-type ideals

Let $R$ be a local Gorenstein ring and let $I\subset R$ be a perfect ideal of Dynkin type. A structure map $w^{(i)}_{j,k}(I)$ has strictly positive rank modulo the maximal ideal if its reduction modulo the maximal ideal of $R$ has positive rank. Dynkin structure-map conjecture. Some structure map $w^{(i)}_{j,k}(I)$, with $i=2,3$ and $j\geq1$, has strictly positive rank modulo the maximal ideal of $R$. The conjecture predicts nontrivial higher structure in every Dynkin-type resolution; the source discusses computed cases and the expected relevance to generic perfect ideals, but gives no resolution status.