Positivity conjecture for the Jacobian of the multigraph moment map

Let $h:\mathbb{R}^n\setminus\{\mathbf{0}\}\to\mathbb{R}^n$ be defined componentwise by

$$h_i(\boldsymbol{\beta})=\sum_{j\ne i}\frac{\beta_i\beta_j}{2\psi-\beta_i\beta_j}.$$

Let $\mathbf{J}$ be the Jacobian of $h$, and let

$$\mathcal{B}=\{\boldsymbol{\beta}\mid \boldsymbol{\beta}>\mathbf{0},\;\boldsymbol{\beta}\boldsymbol{\beta}^{T}>2\psi\mathbf{E}\}.$$

For $\boldsymbol{\beta}\in\mathcal{B}$, let $\lambda_n(\boldsymbol{\beta})$ be the smallest eigenvalue of $\mathbf{J}(\boldsymbol{\beta})$. Positivity conjecture. For all $\boldsymbol{\beta}\in\mathcal{B}$, $\lambda_n(\boldsymbol{\beta})>0$. In particular, $\mathbf{J}$ is positive definite and nonsingular. Furthermore, if $\boldsymbol{\beta}\geq 1$ entrywise, then $\lambda_n(\boldsymbol{\beta})\geq 1$. Numerical evidence suggests this conjecture, which would establish nonsingularity and positive definiteness of the Jacobian on the feasible set.