Asymptotic growth conjecture for symmetric domino tilings of Aztec diamonds

Let $AD(n)$ be the Aztec diamond of order $n$. Let $O(2n,[2n])$ be the set of off-diagonally symmetric domino tilings of $AD(2n)$ with no boundary defect, and let $D(2n-1)$ be the set of nearly off-diagonally symmetric domino tilings of $AD(2n-1)$. Asymptotic growth conjecture. We have

$$\lim_{n\to\infty}|O(2n,[2n])|^{\frac{2}{(2n)^2}}=\lim_{n\to\infty}|D(2n-1)|^{\frac{2}{(2n-1)^2}}=\sqrt{2}.$$

The conjecture concerns the exponential growth rate of these symmetric and nearly symmetric tiling classes. The paper reports verification of both conjectures in its open-problems section up to $n=35$; no proof or resolution of this asymptotic assertion is supplied.