Lerman–Montgomery–Sjamaar conjecture on cotangent symplectic reductions

Let $G$ and $H$ be Lie groups and let $X$ and $Y$ be smooth manifolds on which $G$ and $H$, respectively, act properly. Assume that the orbit spaces $X/G$ and $Y/H$ are diffeomorphic in the sense that there exists a homeomorphism

$$\phi:X/G\rightarrow Y/H$$

such that the pullback map

$$\phi^*:C^\infty(Y/H):=C^\infty(Y)^H\rightarrow C^\infty(X/G):=C^\infty(X)^G$$

is an isomorphism. The cotangent symplectic reductions $T^*X\sslash G$ and $T^*Y\sslash H$ are stratified symplectic spaces, with smooth functions inherited from invariant smooth functions on the corresponding cotangent bundles. Lerman–Montgomery–Sjamaar conjecture. Under these assumptions, $T^*X\sslash G$ and $T^*Y\sslash H$ are isomorphic as stratified symplectic spaces: there exists a homeomorphism between them whose pullback is an isomorphism of their Poisson algebras of smooth functions. The conjecture asserts that the cotangent symplectic reduction is determined by the smooth orbit space. The source attributes this conjecture to an earlier work and presents it without evidence of resolution; its status is therefore open.