Generic 2:2 relation for natural parameters of degenerate tetrahedra

Let $\alpha,\beta,\gamma,\delta,\varsigma\in\mathbb R$ satisfy

$$|\alpha|+|\beta|+|\gamma|+|\delta|>|\alpha+\beta+\gamma+\delta|,\qquad \alpha\beta\gamma\delta>0,$$

and let $\varrho_+(\alpha,\beta,\gamma,\delta)$ denote the unique positive root of the discriminant quadratic described in the surrounding discussion. Generic 2:2-relation conjecture. The equations referenced in the source generically define a $2\!:\!2$ relation between the natural parameters of degenerate tetrahedra and the semi-algebraic set

$$\left\{\alpha,\beta,\gamma,\delta,\varsigma\in\mathbb R\mathrel{\bigm|}|\alpha|+|\beta|+|\gamma|+|\delta|>|\alpha+\beta+\gamma+\delta|,\ \alpha\beta\gamma\delta>0,\ \varsigma>\sqrt{\varrho_+(\alpha,\beta,\gamma,\delta)}\right\}.$$

The conjecture describes the generic multiplicity of the parameter correspondence in the region where the preceding analysis predicts three real roots.