Sparse relaxation order conjecture for finite convergence

Let a polynomial optimization problem have a sparse moment relaxation and the corresponding dense moment relaxation, with relaxation order measured by the parameter in the respective hierarchy. Sparse relaxation order conjecture. If the sparse moment relaxation exhibits finite convergence, then it does so at a relaxation order no larger than the relaxation order at which the dense relaxation is exact. This conjecture asserts that exploiting correlative sparsity does not require a higher relaxation order than the dense formulation when finite convergence occurs; the provided text does not state whether the claim is proved or remains open.