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Rigid non-cohomologically rigid local systems

For any even natural number $r \ge 2$, we construct an irreducible rigid non-cohomologically rigid complex local system of rank $r$ on a smooth projective variety depending on $r$. For $r=2$, we construct an irreducible rigid non-cohomogically rigid local system of rank $2$ on a quasi-projective variety which becomes cohomologically rigid after fixing the conjugacy classes of the monodromies at infinity. v2: We added a remark due to Alexander Petrov: by taking the exterior product of our examples with a (cohomologically) rigid local system with infinite monodromy, we obtain examples of rigid non-cohomologically rigid local systems with infinite monodromy.