Engström's connectivity conjecture for large fiber graphs

Let $G$ be a fiber graph arising from a lattice ideal and a Gröbner basis. A fiber graph is $N$-large if it is the preimage of a monomial $m$ divisible by $(y_1\cdots y_n)^N$; for contingency-table ideals, this means that every row and column sum is at least $N$. Write $\delta(G)$ for the minimum vertex degree and $\kappa(G)$ for the connectivity.

Engström's conjecture. For any lattice ideal with a Gröbner basis, there is an $N$ such that the connectivity of every $N$-large fiber graph satisfies

$$\kappa(G)=\delta(G).$$

The conjecture predicts that sufficiently large fiber graphs have connectivity controlled exactly by their minimum degree. The paper confirms it for a large class of fiber graphs associated with an important class of Gröbner bases; the general statement remains open.