Zariski-density of local systems of geometric origin

Establish that, for any smooth complex variety X and integer r≥1, the locus of local systems of geometric origin is Zariski-dense in the character variety M_B(X,r).

Background

This conjecture extends the ‘rigid implies motivic’ philosophy to all components of the character variety, predicting a pervasive presence of motivic local systems. The paper notes this conjecture would imply the earlier density conjecture for finite mapping class group orbits, and also explains why the conjecture is false in certain settings.