Atiyah–Sutcliffe's multiplicative conjecture for the Atiyah determinant

Let $x_1,\dots,x_n$ be distinct points in $\mathbb{R}^3$, and let $D(x_1,\dots,x_n)$ be the Atiyah determinant; for each $i$, write $D(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)$ for the determinant associated with the configuration obtained by deleting $x_i$. Atiyah–Sutcliffe's multiplicative conjecture. For all such configurations,

$$|D(x_1,\dots,x_n)|^{n-2}\geq \prod_{i=1}^n |D(x_1,\dots,x_{i-1},x_{i+1},\dots,x_n)|.$$

This is the third conjecture formulated by Atiyah and Sutcliffe. The paper discusses evidence and special cases, but no general proof is given.