  Fix a finite set of primes $\mathcal{S}$, for an abelian variety $A_0$ over
$\overline{\mathbb{Q}}$ (with extra structures) which has supersingular good
reductions at all $\ell\in\mathcal{S}$, we consider the simultaneous reduction
modulo $\ell\in\mathcal{S}$ of several copies of $p$-adic/prime-to-$N$ Hecke
orbits of $A_0$. We show that this simultaneous reduction map is always
surjective under certain rationality condition on the copies of the Hecke
orbits.
