Complexity of testing cone balancedness

Determine whether testing whether a rational cone is balanced is Co-NP-complete.

Background

The paper introduces balancedness for rational cones: a cone is balanced if every integer vector in it can be decomposed into two integer vectors in the cone whose componentwise difference has absolute value at most one. Balancedness is central to the paper because the authors characterize, for full-dimensional polyhedra contained in the strictly positive orthant, when an integer equitable subpartitionable description exists in terms of balanced vertex normal cones.

The authors note that recognizing box-TDI polyhedra is co-NP-complete and use this complexity result as motivation for conjecturing that testing balancedness of a cone is also Co-NP-complete. The conjecture is explicitly left unresolved.