Malkoun's singular-value conjecture for Weyl-equivariant maps

Let $X\subset\mathfrak{h}\otimes\mathbb{R}^3$ be the configuration space and let

$$F:X\to\left(\mathbb{C}^{m+1}\setminus\{\mathbf{0}\}\right)^n/T^n$$

be the map whose components are the polynomial representatives defining the Weyl-equivariant map $f$. Let $r_{col}$ be the common rank of $f(\mathbf{x})$ for collinear configurations. Define

$$g=\operatorname{diag}\left(\binom{m}{0}^{-1},\binom{m}{1}^{-1},\ldots,\binom{m}{m}^{-1}\right)$$

and

$$\Delta(\mathbf{x})=s_{r_{col}}\left((\sqrt{g})^{-1}\operatorname{Sing}\left(\sqrt{g}F(\mathbf{x})\right)\right),$$

where $s_j$ is the $j$-th elementary symmetric polynomial in the diagonal entries of the singular-value matrix.

Malkoun's singular-value conjecture. For every $\mathbf{x}\in X$,

$$\Delta(\mathbf{x})\geq\Delta(\mathbf{x}_{col}).$$

This is presented as a quantitative refinement of the rank conjecture: the non-vanishing of $\Delta$ is equivalent to the first conjecture, and the normalization by $g$ makes $\Delta$ $SU(2)$-invariant. The source gives no resolution of this stronger conjecture.