The strong cluster-map conjecture for unipotent cells

Let $Q$ be a finite connected non-Dynkin quiver without loops, let $\Lambda$ be its completed preprojective algebra, let $W$ be the associated Coxeter group, and let $U^w$ be the unipotent cell associated with $w\in W$. Let $\varphi_w:\operatorname{Sub}\nolimits\Lambda/I_w\to\mathbb{C}[U^w]$ be the map induced by the GLS map and restriction to $\mathbb{C}[U^w]$.

Strong cluster-map conjecture. For every $w\in W$, the coordinate ring $\mathbb{C}[U^w]$ is a cluster algebra modelled by the strong cluster map $\varphi_w$ for the standard component of the cluster tilting graph of $\operatorname{Sub}\nolimits\Lambda/I_w$, with all coefficients inverted.

This would extend the known Dynkin examples and identify the categorical cluster structure with the coordinate rings of arbitrary unipotent cells in this setting. The statement is posed as a conjecture and is not resolved in the supplied text.