Compatibility conjecture for Borel and general linear moment-map quotients

Let $B$ be the Borel subgroup of $GL_n(\mathbb{C})$, let $G=GL_n(\mathbb{C})$, and let $\mu_B$ and $\mu_G$ denote the corresponding moment maps. Consider the GIT and affine quotients appearing in the diagram.

Compatibility conjecture. The following diagram commutes:

\xymatrix@-1pc{ \ar@{->>}[dd] \mu_B^{-1}(0)/\!/_{\det}B \ar@{..>}[rr] & & \mu_G^{-1}(0)/\!/_{\det} GL_n(\mathbb{C}) \ar@{->>}[dd] \\ & {\reflectbox{\rotatebox[origin=c]{180}{$\circlearrowleft$}}} & \\ \mu_B^{-1}(0)/\!/B \ar@{..>}[rr] & & \mu_G^{-1}(0)/\!/GL_n(\mathbb{C}) \\ }

The diagram is intended to relate the Borel quotients to the Hilbert-scheme and symmetric-product quotients for $GL_n(\mathbb{C})$. The supplied text gives no resolution status, so the claim remains open here.