Conjecture on the E_8 Weisfeiler graded Lie algebra in characteristic two

Let $p=2$, and let $\mathfrak{g}$ be of type $E_8$ with maximal subalgebra $M_8$ and $M_8$-invariant subspace $L'_{-1}$ as in Theorem none8p2. For the Weisfeiler filtration $\mathcal{F}$ associated with $(L'_{-1},M_8)$, let $\mathcal{G}$ be the corresponding graded Lie algebra and let $\overline{\mathcal{G}}$ denote its quotient by the Weisfeiler radical. The E_8 Weisfeiler conjecture. One has

$$\mathcal{H}(6;\underline{1})\otimes\mathcal{O}(3;\underline{1})\subset\overline{\mathcal{G}}\subseteq\bigl(\operatorname{Der}(\mathcal{H}(6;\underline{1}))\otimes\mathcal{O}(3;\underline{1})\bigr)\rtimes\bigl(1_{\mathcal{H}(6;\underline{1})}\otimes W(3;\underline{1})\bigr).$$

This conjecturally describes the degenerate Weisfeiler filtration in type $E_8$; the source supplies dimension-based remarks but no proof of the asserted inclusions.