The periodic odd-size O(1) loop-model largest-component conjecture

Consider the dense O(1) loop model with periodic boundary conditions and odd system size $L=2n-1$, and normalise its groundstate wavefunction so that its smallest element is $1$. Let $A_{n-1}$ denote the number of $(n-1)\times(n-1)$ alternating sign matrices. The periodic odd-size largest-component conjecture. The largest element of the wavefunction is $A_{n-1}^2$.

The square is related in the source to cyclically symmetric self-complementary plane partitions. The assertion is based on finite-size wavefunctions and remains conjectural.