Bipartite determinant inequality for Gaussian Markov random fields

Let $G=(V,E)$ be a finite bipartite graph. For $x\in(-1,1)$, let $\tau(G,x)=\det(\Sigma(G,x))$, where $\Sigma(G,x)$ is the unique positive-definite matrix with diagonal entries $1$ and edge entries $x$ maximizing the determinant. Bipartite determinant conjecture. For every such graph $G$ and every $x\in(-1,1)$,

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

This is a weaker consequence of the preceding conjecture because $\tau(G,x)$ is even for bipartite graphs. It remains open, although many related examples are known.