Conjecture on the E_7 Weisfeiler graded Lie algebra in characteristic two

Let $p=2$, and let $\mathfrak{g}$ be of type $E_7$ with maximal subalgebra $M_7$ and $M_7$-invariant subspace $L'_{-1}$ as in Theorem none7p2. For the Weisfeiler filtration $\mathcal{F}$ associated with $(L'_{-1},M_7)$, let $\mathcal{G}$ be the corresponding graded Lie algebra and let $\overline{\mathcal{G}}$ denote its quotient by the Weisfeiler radical. The E_7 Weisfeiler conjecture. One has

$$\mathfrak{psl}(4)\otimes\mathcal{O}(3;\underline{1})\subset\overline{\mathcal{G}}\subseteq\bigl(\mathfrak{sp}(6)\otimes\mathcal{O}(3;\underline{1})\bigr)\rtimes\bigl(1_{\mathfrak{psl}_4}\otimes W(3;\underline{1})\bigr).$$

The claim is supported by the paper’s analysis of the degree-zero derivations and dimensions, but remains conjectural in the supplied text.