The Mumford–Tate projection conjecture for J_{p^3}

Let $p$ be an odd prime, let $J_{p^3}$ be the Jacobian of $C_{p^3}\colon y^2=x^{p^3}-1$, and write $J_{p^3}\simeq X_2\times X_1\times J_p$, where $X_2$ is the absolutely simple factor described in the source. The Mumford–Tate projection conjecture for $J_{p^3}$. The projection

$$\operatorname{MT}(J_{p^3})\longrightarrow\operatorname{MT}(X_2)$$

is an isomorphism. The source verifies the analogous statement computationally for odd primes $p\leq13$ and says that the general case remains an open question.