Asymptotic reciprocal-derivative conjecture for Dirichlet and Dedekind zeta-functions

Let $\chi$ be a primitive Dirichlet character, and let $K$ be an Abelian number field. Write $\rho=\beta+i\gamma$ for nontrivial zeros of the relevant Dirichlet $L$-function $L(s,\chi)$ and Dedekind zeta-function $\zeta_K(s)$, respectively. Reciprocal-derivative asymptotic conjecture. As $T\to\infty$, one has

$$\sum_{0<\gamma\le T}\frac{1}{L'(\rho,\chi)}\sim\frac{T}{2\pi},\qquad \sum_{0<\gamma\le T}\frac{1}{\zeta_K'(\rho)}\sim\frac{T}{2\pi}.$$

The conjecture is suggested by Landau–Gonek-type formulas and would sharpen the lower-bound estimates used for summatory Möbius functions in arithmetic progressions and over Abelian number fields. Its resolution is not supplied in the source.