The characteristic-polynomial conjecture for $y^2=x^{2p}-1$

Let $C_{2p}$ be the genus $g$ curve

$$y^2=x^{2p}-1,$$

where $p$ is prime. For the characteristic polynomials $P_{d,j}(T)$ associated with the component representatives indexed by $d$ and $j$, the characteristic-polynomial conjecture.

$$P_{d,j}(T)=(T^g+1)^2$$

for any $d$ relatively prime to $2g$ and $j=0$ or $1$. This is a conjectural pattern for the components of the Sato–Tate group of the second family and is presented as a generalization of the computed examples.