The abstract classification conjecture for hypertoric varieties

A symplectic algebraic variety is a connected algebraic variety equipped with a symplectic structure. Let $T^d$ be an algebraic torus of dimension $d$ acting effectively and hamiltonianly on such a variety, and suppose the variety is projective over its affinization. A hypertoric variety is one obtained by the hypertoric quotient construction. The abstract classification conjecture. Any connected, symplectic, algebraic variety which is projective over its affinization and admits an effective, hamiltonian action of the algebraic torus $T^d$ is equivariantly isomorphic to a Zariski open subset of a hypertoric variety. This would provide an algebraic analogue of Bielawski's classification theorem in hyperkähler geometry; the corresponding algebraic theorem had not been proved in the source.