Bethe perm-vector pseudo-codeword conjecture

Let $\mathcal{C}$ be a binary linear code with parity-check matrix ${\bm{H}}\in{\mathbb{F}}_2^{m\times n}$, where $m<n$. Let $\mathcal{I}({\bm{H}})$ be its column-index set, let $\mathcal{K}({\bm{H}})$ be the fundamental cone associated with ${\bm{H}}$, and let $\mathcal{S}\subseteq\mathcal{I}({\bm{H}})$ have size $m+1$. Define the Bethe perm-vector $\boldsymbol{\omega}$ from $\mathcal{S}$ by $\omega_i=\operatorname{perm}_{\mathrm{B}}({\bm{H}}_{\mathcal{S}\setminus\{i\}})$ for $i\in\mathcal{S}$ and $\omega_i=0$ otherwise. Bethe perm-vector pseudo-codeword conjecture. The vector $\boldsymbol{\omega}$ is a pseudo-codeword of ${\bm{H}}$, namely

$$\boldsymbol{\omega}\in\mathcal{K}({\bm{H}}).$$

The source reports that a proof was subsequently presented by Smarandache, so this conjecture is solved.