Stable-vector embedding conjecture for dual Vinberg representations of type $F_4$

Let $\mathcal{G}$ be a group of type $F_4$, let $x_0$ be the point used to define the dual Vinberg representations, let $h$ be the Coxeter number of $\mathcal{G}$, and for each positive integer $m$ set

$$x_m=x_0+\frac{1}{m}\check\rho.$$

For $m\mid h$, let

$$\iota_m:\check{\mathcal{V}}_{x_h}\hookrightarrow\check{\mathcal{V}}_{x_m}$$

be the natural embedding, and let $\mathfrak f$ be the residue field with characteristic $\operatorname{char}(\mathfrak f)$. Stable-vector embedding conjecture. If $v\in\check{\mathcal{V}}_{x_h}$ is stable for the action of $\mathcal{G}_{x_h}$, then $\iota_m(v)$ is stable for the action of $\mathcal{G}_{x_m}$ if and only if

$$\operatorname{char}(\mathfrak f)\nmid\frac{h}{m}.$$

The conjecture predicts exactly when stability survives the natural embedding from the representation at $x_h$ to that at $x_m$; the paper reports that all known examples satisfy it, while small-characteristic failures of stability can occur and a general proof is not supplied.