Conjecture on dilating positive-weight actions on transversal symplectic slices

Let $Y$ be a symplectic singularity, let $y\in Y$ be a closed point, and let $Y_{y,0}$ be the transversal symplectic slice in the formal product decomposition

$$\widehat{Y}_y=Y_{y,0}\widehat{\times}\widehat{Y_i}_y.$$

A $\mathbb{G}_m$-action is dilating if it scales the symplectic form with positive weight, and positive-weight if every finite-dimensional equivariant subquotient of $H^0(Y,\operatorname{\cal O}_Y)$ has only nonnegative weights. Conjecture on transversal slices. Every transversal slice $Y_{y,0}$ admits a dilating positive-weight $\mathbb{G}_m$-action for which $y$ is its only fixed point.

The conjecture seeks an algebraic enhancement of the formal Weinstein decomposition, providing a contracting symmetry on every transversal slice. The source gives no evidence of a resolution.