Cubic residue asymptotic ordered by the first successive minimum

Let $E$ range over totally real cubic number fields, and let $m_1(E)$ denote the first successive minimum of the positive definite quadratic form $Q_E$ on $\mathcal{O}_E/\mathbb Z$. Cubic residue asymptotic. There exists a constant $\alpha_3>0$ such that, as $X\to\infty$,

$$\sum_{E:~m_1(E)\leq X}\mathop{\rm res}_{s=1}\zeta_E(s)\sim \alpha_3X^{\frac{5}{2}},$$

where the sum extends over all totally real cubic number fields $E$ for which $m_1(E)\leq X$. This conjectures the precise nonzero asymptotic suggested by the proved upper and lower bounds for cubic fields; the required uniformity in the underlying test-function estimates is not established.