Derived equivalence between Shioda's fourfold and CM Mumford fourfold

Shioda proved that the Jacobian $A_S$ of the curve $y^2 = x^9 -1$ is a 4-dimensional CM abelian variety with codimension 2 Hodge cycles not generated by divisors. It was noted by Shioda that this behavior resembles the abelian varieties constructed by Mumford. We prove that Shioda's fourfold $A_S$ is isogenous to an abelian variety that is derived equivalent to a CM Mumford fourfold $A_M$, but it cannot be realized as a special case of Mumford's construction.