Positivity conjecture for simple VOAs generated by sigma-type Ising vectors

Let $V$ be a simple VOA satisfying the paper's Conditions 1, let $E_V$ be the set of Ising vectors of $V$ of $\sigma$-type, and let $G_V=\langle \sigma_e\mid e\in E_V\rangle$. The real Matsuo algebra associated with $G_V$ is denoted by $B_{1/2,1/2}(G_V)_{\mathbb{R}}$. Positivity conjecture. The bilinear form on the $\mathbb{R}$-span of $E_V$ is positive definite; equivalently, the non-degenerate quotient of $B_{1/2,1/2}(G_V)_{\mathbb{R}}$ is positive definite. This positivity would imply, by the preceding corollary, that the real VOA generated by $E_V$ is a compact real form and that the relevant 3-transposition groups occur in Matsuo's list. The source gives no resolution.