Somoza's endomorphism-ring conjecture for a cyclic plane quintic

Let $C$ be the smooth plane curve

$$C:y^5=x^4-24x^3+3x^2+x.$$

Let $J$ be its Jacobian, let $K=\mathbb{Q}(\zeta_9)^+(\zeta_5)$, and let $\mathcal{O}_K$ denote the ring of integers of $K$. Somoza's endomorphism-ring conjecture. The Jacobian $J$ satisfies

$$\operatorname{End}(J_{\overline{\mathbb{Q}}})\cong\mathcal{O}_K.$$

This conjecture predicts complex multiplication by the full ring of integers of the indicated number field for the Jacobian of Somoza's cyclic plane quintic. The supplied text gives no evidence of a resolution.