Structural decomposition conjecture for local systems with restricted variation

Let $G$ be a reductive group and let $D^{\circ}$ be the formal punctured disc. Let $\operatorname{LS}_G^{\mathrm{restr}}(D^{\circ})$ denote the stack of $G$-local systems with restricted variation, let $\operatorname{LS}_G(D^{\circ})^{\mathrm{et}}$ be the étale version of the stack of $G$-local systems, and let $(\operatorname{LS}_G(D^{\circ})^{\mathrm{et}})^{\mathrm{red}}$ denote its reduced substack.

Structural decomposition conjecture. The stack $\operatorname{LS}_G^{\mathrm{restr}}(D^{\circ})$ is a disjoint union of formal completions of a collection of pairwise non-intersecting closed substacks of

$$(\operatorname{LS}_G(D^{\circ})^{\mathrm{et}})^{\mathrm{red}}.$$

The source identifies this as a structural result that is not known yet. Such a decomposition would clarify the geometry of the moduli stack of local systems with restricted variation.