The singular reduction conjecture for Hamiltonian circle actions

Let $X$ be a Liouville manifold equipped with a Hamiltonian $S^1$ action such that the fixed locus $X^{S^1}$ has codimension four and there are no finite non-trivial stabiliser groups. Let $\mu:X\to\mathbb{R}$ be the moment map, let $a\in\mathbb{R}$ be a singular value in the interior of $\operatorname{Im}\mu$, and let $U$ be the smooth locus in $\mu^{-1}(a)$. Singular reduction conjecture. There is a quasi-equivalence

$$\mathcal{W}_{S^1}(X)_{-e^a}\cong\mathcal{W}(U/S^1).$$

This proposes that at singular moment-map values the relevant equivariant Fukaya category is governed by the quotient of the smooth part of the singular fibre, rather than by ordinary symplectic reduction. The source presents this as a general conclusion under the stated codimension and stabiliser assumptions.