Graver-basis edge-connectivity conjecture for fiber graphs

Let $A\in\mathbb{Z}^{d\times n}$ be an integer matrix with $\ker(A)\cap\mathbb{Z}_{\geq 0}^n=\{\mathbf{0}_n\}$. For each right-hand side, let the Graver fiber graph be the fiber graph whose edges are defined by the Graver basis of $A$; its edge-connectivity is the minimum number of edges whose removal disconnects the graph, and its minimal degree is the minimum vertex degree.

Graver-basis edge-connectivity conjecture. In all Graver fiber graphs of $A$, the edge-connectivity equals the minimal degree.

The source motivates this conjecture from results showing that Graver moves give strong edge-connectivity properties, while noting that the corresponding vertex-connectivity question is unresolved. The supplied text gives no resolution of this edge-connectivity conjecture.