The real-moment limit conjecture for ideal class counting functions

Let $a(\beta,\beta')$ denote the quantities appearing in Theorem, and let $\beta>0$ be real. For $\beta$ integral or half-integral, the corresponding $\beta$-moment has matching lower and upper asymptotic bounds, and the limiting quantities do not depend on the ideal class $\sigma\in N$. Real-moment limit conjecture. For real $\beta>0$, the $\liminf$ and $\limsup$ in Theorem are equal, and they are independent of $\sigma\in N$. This would extend the equality already observed when $\beta\in\mathbb{Z}/2$ to all positive real moments.