Conjecture on the smallest orbit of stable vectors for stable gradings with s_0=0

Let $\theta$ be a stable grading on the dual Lie algebra $\widehat{\mathfrak g}$, with associated parahoric subgroup $P$. Let $p:\widehat{\mathfrak g}_1\to\widehat{\mathfrak g}(1)$ be the projection, let $\widehat{G}(0)$ act on $\widehat{\mathfrak g}(1)$, and let $\mathcal{O}_P$ be an orbit to be specified. Write $\mathfrak u_{\widehat P}$ for the relevant unipotent subgroup. Smallest-orbit conjecture. (i) There is a $\widehat{G}(0)$-orbit $\mathcal{O}_P\subset p(\widehat{\mathfrak g}_1^{\operatorname{st}})$ satisfying $\exp(\mathcal{O}_P)\subset\mathfrak u_{\widehat P}$. (ii) For any $X\in\widehat{\mathfrak g}_1^{\operatorname{st}}$, there exists $g\in\exp(\widehat{\mathfrak g}(m))\subset\widehat{G}_0$ such that $p(\operatorname{Ad}_gX)\in\mathcal{O}_P$. The conjecture is proposed as a description of the smallest orbit in the image when the Kac coordinate $s_0=0$, generalizing the stated monodromy proposition; the source supplies no resolution.