The simple-factor decomposition conjecture for Jacobians of Fermat-type curves

Let $m$ be a positive odd integer, and let $J_m$ denote the Jacobian associated with the curve $C_m\colon y^2=x^m-1$. Write $\phi(d)$ for Euler's totient function. The simple-factor decomposition conjecture. The decomposition of $J_m$ into simple factors contains exactly one simple factor of dimension $\phi(d)/2$ for each divisor $d$ of $m$. Moreover, if $p$ is a prime divisor of $m$, the factor of dimension $\phi(p)/2$ is the Jacobian of $C_p\colon y^2=x^p-1$. The conjecture is motivated by computed Frobenius polynomials, and the source gives no general proof or resolution.