Characterization of supports of variable-wise log-concave Laurent polynomials

Characterize exactly which finite saturated subsets of the integer lattice can occur as the support of a Laurent polynomial in the variable-wise log-concavity class $\LC$.

Background

The paper proves that every Laurent polynomial in the class $\LC$ has a finite saturated support. However, the converse fails: the paper gives a saturated set, $\{(0,0),(1,0),(2,1)\}$, for which no Laurent polynomial with positive coefficients and that support belongs to $\LC$.

Consequently, determining a necessary and sufficient condition for a finite saturated set to arise as the support of an $\LC$ Laurent polynomial remains unresolved.