Characterization of representable polynomials by factorization modulo square ideals

Let ${\mathbb F}$ be a field of characteristic $2$, let $P\in{\mathbb F}[x_1,\dots,x_m]$, and let $\ell\in{\mathbb F}^m$ be a tuple of squares. Write $\operatorname{Mult}_{\xi}P$ for the polynomial construction used in the source and $\mathcal{I}(\ell)$ for the corresponding ideal. The polynomial $P$ is representable if it has a symmetric determinantal representation over ${\mathbb F}$ with entries in ${\mathbb F}\cup\{x_1,\dots,x_m\}$.

Factorization characterization conjecture. The polynomial $P$ is representable if and only if, for some (equivalently any) tuple of squares $\ell\in{\mathbb F}^m$, $\operatorname{Mult}_{\xi}P$ is factorizable modulo $\mathcal{I}(\ell)$ into linear polynomials

$$L_1,\dots,L_k\in{\mathbb F}[\xi_1,\dots,\xi_m][x_1,\dots,x_m].$$

The paper establishes several results on symmetric determinantal representations in characteristic $2$, including a complete characterization for multilinear polynomials, but identifies a full characterization of representable polynomials as the main remaining open question. The notation for $\operatorname{Mult}_{\xi}$ and $\mathcal{I}(\ell)$ is not defined in the supplied context.