Delzant's conjecture for multiplicity-free Hamiltonian manifolds

Let $K$ be a connected compact Lie group, let $M$ be a multiplicity-free compact Hamiltonian $K$-manifold with moment map $\mu:M\rightarrow\mathfrak k$, and fix a maximal torus $T_K\subset K$ with Lie algebra $\mathfrak t_K$ and a Weyl chamber $C\subset\mathfrak t_K$. Define the moment polytope $\Delta(M)=\mu(M)\cap C$ and let $K(M)$ be the stabilizer of a general point of $\mu^{-1}(C)$. Delzant's conjecture. If $M_1$ and $M_2$ are multiplicity-free compact Hamiltonian $K$-manifolds such that $\Delta(M_1)=\Delta(M_2)$ and $K(M_1)=K(M_2)$, then $M_1$ and $M_2$ are $K$-equivariantly symplectomorphic. This conjecture asserts that the moment polytope and principal isotropy group are complete invariants for multiplicity-free compact Hamiltonian actions; the source presents it as an application motivated by the spherical-variety classification theorem, without stating its resolution.