Flatness conjecture for the Borel Zastava projection

Let $\fg$ be a Kac–Moody algebra, $\fb$ a Borel subgroup, $\bC$ the curve, and $\mu$ a dominant coweight. Let $\on{Maps}^\mu(\bC,\CG_{\fg,\fb})$ be the corresponding space of maps, and let

$$\varrho^\mu_\fb:\on{Maps}^\mu(\bC,\CG_{\fg,\fb})\to \overset{\circ}{\bC}{}^\mu$$

be its projection to the configuration space of colored divisors. Flatness conjecture. The projection $\varrho^\mu_\fb$ is flat. Equivalently, for any point $\bc\in \overset{\circ}\bC$, the preimage of $\mu\cdot \bc\in \overset{\circ}{\bC}{}^\mu$ is equidimensional of dimension $|\mu|$. The conjecture is stated immediately after the dimension calculation for the Borel Zastava space; the source says it will be established when $\fg$ is of finite and affine type, leaving the general case unresolved.