Equivariant local-structure conjecture for non-Borel extra-dimension-six ideals

Let $I_{\lambda}$ be a non-Borel monomial ideal of $\Bbbk[X_1,X_2,X_3]$ with extra dimension $6$. Let $\widehat{G}(2,6)$ be the affine cone over the Grassmannian $G(2,6)$, and call a torus action on it standard if it is induced by an action on the tangent space at the vertex preserving the Plücker ideal. Then there is a $T$-stable open subset $U$ of $0\in\operatorname{Spec}(\mathbb{A}_{\lambda})$ and an equivariant open immersion

$$f:U\hookrightarrow\widehat{G}(2,6)\times\mathbb{A}^9,$$

where $\widehat{G}(2,6)$ is equipped with a standard $T$-action.

This conjecture would provide the local model needed to compute equivariant Hilbert functions for the relevant non-Borel ideals. It is used conditionally in the paper and remains open.