Before revision:  We show further that the improved upper Fr\'echet--Hoeffding bound is not the dual optimizer of this transport problem . In turn we prove that the improved upper bound is the dual optimizer of a relaxed version of the initial transport problem, thus proving sharpness of the improved upper Fr\'echet--Hoeffding bound for the relaxed Fr\'echet class.
After revision:  We show further that the improved upper Fr\'echet--Hoeffding bounds provide ad-hoc upper bounds for these Fr\'echet class.