The negativity-locus characterization of compact torus orbit spaces

Let $\mathbb{T}^n$ be a compact torus, let $\mathcal{G}$ act on it, and let $\mathcal{T}$ denote the image of $\mathbb{T}^n$ under the orbit-map parametrization $\vartheta=(\theta_1,\ldots,\theta_m)$. Let $M einmathbb{Q}[z]^{m\times m}$ be a matrix polynomial such that

$$\tilde{M}=M(\theta_1,\ldots,\theta_m).$$

Let $\mathcal{I}\subseteqmathbb{Q}[z]$ be the ideal of relations among $\theta_1,\ldots,\theta_m$, and let $\mathcal{V}(\mathcal{I})\subseteqmathbb{C}^m$ be its variety. The negativity-locus conjecture. If $z einmathcal{V}(\mathcal{I})$ and $M(z)$ is Hermitian negative semi-definite, then $z einmathcal{T}$. Equivalently, the $\mathbb{T}$-orbit space is characterized by the variety of the relations together with the negative-semidefiniteness of $M(z)$. This is proposed as a complex analogue of the Procesi–Schwarz characterization of real orbit spaces for linear compact Lie-group actions; the supplied text does not indicate whether the conjecture has been resolved.