Nonzero-coefficient conjecture for the Combinatorial Nullstellensatz framework

Let $\mathcal H$ be an $n$-uniform linear hypergraph with $n$ edges. For each pair of indices $1\leq i<k\leq n$, let $\Phi_{i,k}(\mathbf{x})$ be the polynomial defined from the chosen identifier spanning trees, and let a Vandermonde-completable monomial mean a monomial corresponding to an orientation whose base cliques can be oriented as transitive tournaments with every total in-degree at most $n-1$.

Nonzero-coefficient conjecture. One can choose the identifier spanning trees in a way that there is a Vandermonde-completable monomial with nonzero coefficient in

$$\prod_{i<k}\Phi_{i,k}(\mathbf{x}).$$

The conjecture is the missing algebraic step in the paper's approach to the Erdős–Faber–Lovász conjecture. The corresponding existence of Vandermonde-completable orientations is proved, but the required nonvanishing of the coefficient sum is left open.