BdGN's largest-entry conjecture for the periodic model

Let

be the homogeneous-limit ground-state eigenvector of the periodic transfer matrix at $q=e^{2i/3}$, normalized so that its smallest entry $_{_0}$ is $1$, and let $A(n)$ denote the number of Alternating Sign Matrices of size $n$. **BdGN's largest-entry conjecture.** The largest entry of

, corresponding to the link pattern with arches connecting consecutive points, is $A(n-1)$. This refines the known result that the sum of entries is $A(n)$; the largest-entry assertion remains unproved in the supplied text.