Conjectural mean-size asymptotic for primitive odd class-group components

Let $k\geq 3$ be odd, let $H_k(-d)^*$ denote the relevant primitive component of the class group, and let $\phi\in C_c^\infty(\mathbb{R}^+)$. Write $\hat{\phi}$ for the Mellin transform of $\phi$, and let $C_{1,k}$ be the constant from the paper's general asymptotic theorem. The smoothed aggregate size is

Mean-size conjecture.

$$\sum_d^* \phi\left(\frac{d}{D}\right)|H_k(-d)^*|=\frac{2}{\pi^2}\hat{\phi}(1)D+C_{1,k}\hat{\phi}\left(\frac{1}{2}+\frac{1}{k}\right)D^{\frac{1}{2}+\frac{1}{k}}+o\left(D^{\frac{1}{2}+\frac{1}{k}}\right).$$

This prediction is motivated by assuming sufficiently strong equidistribution for all odd $k\geq 3$, so that the absence of Heegner points above height $D^{\frac12-\frac1k}$ is detectable. It extends the known $k=3$ situation connected with the Davenport--Heilbronn asymptotic; the source presents the general statement as conjectural.