Negativity conjecture for the Jensen-Shannon and Kullback-Leibler divergence Jacobian

Let $\Omega^{\mathrm{o}}=\{(\mu,\nu)\in(0,1)\times(0,1)\mid \nu<\mu\}$ be the interior of $\Omega$, and let $J$ be the Jacobian defined in Eq.. Negativity conjecture. For every $(\mu,\nu)\in\Omega^{\mathrm{o}}$,

$$\det(J(\mu,\nu))<0.$$

The claim is supported in the paper by a visualization showing the negative of the determinant as positive throughout the stated domain, but no proof or resolution is supplied in the provided text.