Uniqueness of variational limits under measured and pointed Gromov–Hausdorff convergence

Let $\mathcal{M}$ be the family of source spaces and $\mathcal{Y}$ the family of pointed target spaces. Suppose that a net $M_i\in\mathcal{M}$ converges in the measured Gromov–Hausdorff topology to a measured metric space $M$, and that a net $(Y_i,y_i)\in\mathcal{Y}$ converges in the pointed Gromov–Hausdorff topology to a proper metric space $(Y,y)$. Uniqueness conjecture. The limit of the energy functionals $E_{M_i,Y_i}$ should be uniquely determined by $M$ and $(Y,y)$. This would make variational convergence depend only on the limiting measured source space and pointed target space, complementing the compactness and convergence results established for the relevant families of energy functionals.