Factorization conjecture for the determinant of the Lyapunov Jacobian

Let $\Sigma$ be a symmetric covariance matrix, let $E$ denote the edge set of the underlying graph, and let $A(\Sigma)_{\cdot,E}$ be the corresponding Jacobian submatrix. Let $H(\Sigma)_{E^c,\cdot}$ denote the restricted kernel matrix on the complement $E^c$ of $E$. The notation is as in the paper.

Factorization conjecture. The determinant of $A(\Sigma)_{\cdot,E}$ factorizes as

$$\det(A(\Sigma)_{\cdot,E}) = 2^p\det(\Sigma) \det\left(H(\Sigma)_{E^c,\cdot}\right).$$

This conjecture would identify the determinant of the restricted kernel as the critical factor, up to the scalar factor $2^p$, and complete the preceding corollary. The source reports substantial numerical evidence but no general proof for $p>3$; the claim that the determinant of $\Sigma$ does not further factor the restricted-kernel determinant remains unproved.