Roverato's conjecture on G-Wishart normalising constants

Let $\mathcal G$ be a graph with $n$ vertices. For $D\in\mathbb S^n_{++}$, let $D^{\mathcal G}$ denote the positive-definite completion of $D$ with respect to $\mathcal G$, and let $\operatorname{Iss}_{\mathcal G}(D^{\mathcal G})$ be the corresponding Isserlis matrix. Write $\mathscr C_{\mathcal G}(\delta,D)$ for the $\mathcal G$-Wishart normalising constant and let $I_n$ be the $n\times n$ identity matrix.

Roverato's conjecture. For every real number $\delta>0$,

$$\mathscr C_{\mathcal G}(\delta,D)=2^{\frac n2}\det\!\left(\operatorname{Iss}_{\mathcal G}(D^{\mathcal G})\right)^{-\frac12}\det(D^{\mathcal G})^{-\frac{\delta-2}{2}}\mathscr C_{\mathcal G}(\delta,I_n).$$

The conjecture would reduce the evaluation of the normalising constant for a general scale matrix to the identity-scale case, with the remaining factors obtainable from positive-definite completion and determinant calculations. It is known for chordal graphs, but the source presents it as unresolved in general.