On irreducibility of six-dimensional compatible systems of $\mathbb{Q}$

We study the irreducibility of 6-dimensional strictly compatible systems of Q with distinct Hodge-Tate weights. We prove that if one of the representations $\rho$ in such a system is irreducible and satisfies a self-dual condition $\rho^{\vee}\otimes\chi\cong\rho$ for some character $\chi$, then all but finitely many of them are irreducible.