Logarithmic concavity conjecture for degree-constrained spanning subgraph coefficients

Let $G=(V,E)$ be a finite graph, and let $f,g:V\rightarrow\mathbb{N}$ be any two functions. Fix the vertex activities $\mathbf{u}$ as in $(4.1)$. Let $N_j$ denote the coefficients of $Z(G,\boldsymbol{1},\mathbf{u};y^{1/2}\boldsymbol{1})$. A sequence has no internal zeros when $N_iN_k\neq 0$ implies $N_j\neq 0$ for every $i\leq j\leq k$, and is logarithmically concave when $N_j^2\geq N_{j-1}N_{j+1}$ for every $j$. The logarithmic concavity conjecture. The sequence $(N_j)$ is logarithmically concave with no internal zeros. The conjecture would give useful structural information about the coefficients of the generating polynomial for degree-constrained spanning subgraphs and could support approximation results for this coefficient sequence. The surrounding discussion notes that the conclusion follows from sufficiently strong sector-nonvanishing hypotheses, but those hypotheses are unreasonably strong; the general assertion remains unresolved here.