Baker–Montgomery conjecture on real zeros of quadratic Dirichlet $L$-function derivatives

Let $d53d_d(d70e_1,d70e_2)$ be the number of real zeros of $L'(s,\chi_d)$ on the interval $[d70e_1,d70e_2]$, where $\chi_d$ is the primitive quadratic character attached to the fundamental discriminant $d$. Baker–Montgomery conjecture. For almost all fundamental discriminants $d$, we have

$$R_d\left(\frac12,1\right)\asymp\log\log|d|.$$

This conjecture concerns the real zeros of derivatives of quadratic Dirichlet $L$-functions and is motivated by the connection with zeros of Fekete polynomials and sign changes of logarithmic derivatives. The paper proves a lower bound within a factor involving an iterated logarithm, but the conjectured order of magnitude remains open.