Pauling's residual entropy conjecture for regular graphs of growing degree

Let $G=G(n)$ be a sequence of $d$-regular graphs of order $n$, with even degree $d=d(n)$. Let $\rho(G)$ denote the residual entropy,

$$\rho(G):=\frac{1}{n}\log\operatorname{EO}(G),$$

where $\operatorname{EO}(G)$ is the number of Eulerian orientations of $G$, and let

$$\widehat\rho(G):=\log\binom{d}{d/2}-\frac{d}{2}\log 2$$

be Pauling's residual entropy estimate. Pauling's residual entropy conjecture. If $d(n)\to\infty$ as $n\to\infty$, then

$$\rho(G)=\widehat\rho(G)+o(1).$$

The conjecture asserts that Pauling's independence heuristic is asymptotically accurate for every sequence of even-degree regular graphs whose degree grows without bound. The paper addresses this conjecture; its status is not resolved by the supplied text.