Conjectured error term for class numbers of orders in quartic fields

Let $S$ be a finite, non-empty set of prime numbers with an even number of elements, and let $\tilde{\beta}_S$ and $\tilde{\beta}$ be as in Theorem 1. For $x>0$, let $\tilde{\beta}_S(x)$ denote the associated counting function $\beta_S(x)$, and define

$$L(x)=\int_1^x\frac{e^t}{t}\,dt.$$

Class-number asymptotic conjecture. Under the conditions of Theorem 1, as $x\to\infty$ one has

$$\pi_S(x)=\frac{1}{2}L(4x)+O\left(\frac{e^{3x}}{x}\right).$$

The paper proves the leading asymptotic $\pi_S(x)\sim e^{4x}/(2x)$ but explains that its method does not preserve the sharper error term available from the Prime Geodesic Theorem. This conjecture predicts an error of order $e^{3x}/x$ for the class-number counting function.