Before revision:  To this end, we establish a dual representation of a constrained transport problem over distributions in the aforementioned Fr\'echet class and show that the improved upper Fr\'echet--Hoeffding bound belongs to the class of admissible functions for the dual optimization problem. The proof of strong duality is based on a general representation result for increasing convex functionals and the explicit computation of the conjugates.
After revision:  To this end, we establish a dual representation of a constrained transport problem over distributions in the aforementioned Fr\'echet classes that extend previous results in the related literature. These proofs are based on representation results for increasing convex functionals and the explicit computation of the conjugates.