Descent of the chord-diagram star product for simple groups

Let $\mathcal{M}^{G}$ be the moduli space associated with a simple group $G$, and let the star product be the product defined by equation on the chord-diagram algebra. Descent conjecture. The star product descends to a star product on $\mathcal{M}^{G}$ for any simple group $G$. The preceding theorem establishes descent for $G=GL(n,\mathbb{C})$ and $G=SL(n,\mathbb{C})$; the assertion extends this to arbitrary simple groups.