VGIT-ring conjecture for hyperkähler Kirwan images

Let $G$ act linearly on $M=T^\ast X$ as above, and assume that the restriction of the real moment map to $X$ is proper. Let $\kappa_{\mathrm{HK}}$ denote the hyperkähler Kirwan map, and let the VGIT ring be the quotient $R^W/I$ defined by the annihilator of the module generated by VGIT residues.

VGIT-ring conjecture. Under these assumptions, the image of the hyperkähler Kirwan map is equal to the VGIT ring:

$$\operatorname{im}(\kappa_{\mathrm{HK}})=R_{\mathrm{VGIT}}.$$

The preceding discussion gives a natural inclusion of the hyperkähler Kirwan kernel in the VGIT ideal and hence a surjection from the hyperkähler Kirwan image onto the VGIT ring. The conjecture asserts that this comparison is an equality; the supplied text gives no resolution.