Half a century ago Manin proved a uniform version of Serre’s celebrated result on the openness of the Galois image in the automorphisms of the $\ell$-adic Tate module of any non-CM elliptic curve over a given number field. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension. Namely, we establish a uniform irreducibility of Galois acting on the $\ell$-primary part of principally polarized Abelian $3$-folds of Picard type without CM factors, under some technical condition which is void in the semi-stable case. A key part of the argument is representation theoretic and relies on known cases of the Gan-Gross-Prasad Conjectures.