Equivariant Grothendieck-resolution embedding for steep partitions

Let $\lambda\neq\stair$ be a steep partition. Let $\mathfrak{X}$ be the flat $SL_2$-equivariant family over $\mathbb{A}^1$ whose special fiber is $\operatorname{Hilb}^n\mathbb{C}^2$, and let $\widetilde{\mathfrak{sl}}_2$ be Grothendieck's simultaneous resolution, with $\varpi:\widetilde{\mathfrak{sl}}_2\to\mathbb{A}^1$ the natural map. The point $y_\lambda$ denotes the corresponding $B$-fixed point in the special fiber.

Embedding conjecture. There exists an $SL_2$-equivariant embedding

$$\widetilde{\mathfrak{sl}}_2\hookrightarrow\mathfrak{X}$$

sending the $B$-fixed point $[1:0]\in\mathbb{P}^1\subset\widetilde{\mathfrak{sl}}_2$ to $y_\lambda$, such that

$$\begin{CD} \widetilde{\mathfrak{sl}}_2 @>>> \mathfrak{X}\\ @V{\varpi}VV @VV{p}V\\ \mathbb{A}^1 @= \mathbb{A}^1 \end{CD}$$

commutes.

This asserts that the Grothendieck simultaneous resolution occurs as an $SL_2$-equivariant subfamily of $\mathfrak{X}$ through the fixed point indexed by $\lambda$. The supplied context does not state whether the claim is proved or remains open.