Asymptotic scarcity conjecture for self-converse digraphs

Let $\mathcal{SC}_n$ be the number of self-converse digraphs on $n$ vertices, and let $\mathcal{D}_n$ be the number of digraphs on $n$ vertices. Asymptotic scarcity conjecture.

$$\frac{\mathcal{SC}_n}{\mathcal{D}_n} \to 0 \text{ as } n\to\infty.$$

The claim gives numerical evidence for the scarcity of self-converse digraphs, which are necessary for digraphs that are strongly determined by the Hermitian spectrum. The source provides numerical evidence rather than a proof, so the asymptotic assertion remains open.