Determinant inequality for Gaussian Markov random fields

Let $G=(V,E)$ be a finite graph. For $x otin(-1,1)$? [sic] the paper defines $\tau(G,x)=\det(\Sigma(G,x))$, where $\Sigma(G,x)$ is the unique positive-definite matrix with diagonal entries $1$ and edge entries $x$ maximizing the determinant. Determinant conjecture. For every graph $G$ and every $x\in[0,1)$,

$$\tau(G,x)\geq(1-x^2)^{|E(G)|}.$$

The inequality is verified in the paper for complete graphs and cycles, and a local version holds for $x$ in a sufficiently small interval; the general conjecture remains open.