Equitable subpartitionability (ESP) is a sufficient condition for an integral system to be box-totally dual integral. We show that the converse fails even when alternative descriptions of the same polyhedron are allowed. We provide a geometrical characterization of polyhedra that can be described by ESP systems in terms of normal cones. We construct a four-dimensional integer polytope described by a totally equimodular matrix that admits no integral ESP description. This answers negatively a previously posed question by the same authors.

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