Flow-closedness conjecture for the norm-square of a complex moment map

Let $K$ be a compact group acting linearly on a vector space, and let $\mu_\mathbb{C}$ be an associated complex moment map. The function $|\mu_\mathbb{C}|^2$ is flow-closed.

Flow-closedness conjecture. If $\mu_\mathbb{C}$ is a complex moment map associated to a linear action by a compact group $K$ on a vector space, then $|\mu_\mathbb{C}|^2$ is flow-closed.

This conjecture would reduce the question of Kirwan surjectivity for hyperkähler quotients to the established surjectivity criterion for $|\mu_\mathbb{C}|^2$ when the function is flow-closed. The source does not state whether the conjecture is resolved.