The field-of-definition conjecture for the zeros of $h_\chi$

Let $N$ be a prime level, let $\chi$ be the character used to define $h_\chi$, and let $B_{m,\chi}$ denote the $m$-th generalised Bernoulli number attached to $\chi$. Write

$$k=\mathbb{Q}(\sqrt{N}).$$

For each zero of $h_\chi$, consider its field of definition. Field-of-definition conjecture. The field of definition of each zero of $h_\chi$ is either $\mathbb{Q}$ or an extension $F$ of the real quadratic field $k$ of degree

$$n=\frac{1}{2}B_{2,\chi}$$

if $N\equiv 1\pmod{8}$, and of degree

$$n=\frac{1}{2}B_{2,\chi}-1$$

if $N\equiv 5\pmod{8}$. This conjecture is motivated by computations of the polynomials $p_N(T)$ and concerns the arithmetic fields generated by the zeros of $h_\chi$; the supplied text gives no resolution, so its status remains open.