My question is about Simpson's motivicity conjecture, that is the conjecture that for any (cohomogically) rigid irreducible connection $(M,\nabla)$ on a smooth complex scheme $X$ is of geometric origin in the sense that there exists $Y\overset{f}{\to}X$ such that $(M,\nabla)$ is a subquotient of $R^nf\_\*\mathcal{O}\_Y$ with the Gauss-Manin connection.


My main question can be put over the top as "why should we believe it?" I'm aware that predictions of this conjecture have been proved, for instance by H.Ésnault and her colaborators. I'd be more interested in a "plausability" sort of criterion. For instance, there are some deformation theoretic arguments explaining why the Fontaine-Mazur conjecture should hold (I'm choosing F-M because it is of a very similar "shape"). Considering this came from Simpson's work on non-abelian Hodge theory, presumably there is a non-abelian Hodge theoretic reason to expect the validity of this conjecture...


Also, how "sharp" do we expect this conjecture to be? Do we know that subquotients of Gauss-Manin connections are rigid?



Simpson's motivicity conjecture
At the time Simpson formulated his conjecture, he had proved that rigid local systems correspond to rational (in a suitable sense) variations of Hodge structure. And, as you point out, we now know that they are integral (Esnault-Groechenig). So rigid local systems have many of the earmarks of geometric local systems.


As for your other questions, geometric local systems, or their subquotients (which would be rationally summands), need not be rigid.


