Flag Hilbert scheme conjecture for the Borel moment-map quotients

Let $B$ be the Borel subgroup of $GL_n(\mathbb{C})$, let $\mu_B$ be the Borel moment map, and write $\mu_B^{-1}(0)/\!/B$ for the affine quotient and $\mu_B^{-1}(0)/\!/_{\det}B$ and $\mu_B^{-1}(0)/\!/_{\det^{-1}}B$ for the GIT quotients associated with the indicated characters. Let the flag Hilbert scheme on a complex plane be the moduli space referred to in the source.

Flag Hilbert scheme conjecture. The following hold:

$$\mu_B^{-1}(0)/\!/_{\det}B\cong\mu_B^{-1}(0)/\!/_{\det^{-1}}B.$$

2. There is a resolution of singularities

$$\mu_B^{-1}(0)/\!/_{\det}B\twoheadrightarrow\mu_B^{-1}(0)/\!/B.$$

3. $\mu_B^{-1}(0)/\!/_{\det}B$ is isomorphic to the flag Hilbert scheme on a complex plane.

The first assertion is identified in the source with variational GIT, or wall-crossing. The proposed resolution and the identification with the flag Hilbert scheme remain conjectural in the supplied text.