Density of finite Mod_{g,n}-orbits on Y(g,n,r)

Establish that for all integers g,n,r, the set of conjugacy classes of representations of the surface group π1(Σ_{g,n}) whose orbits under the mapping class group Mod_{g,n} are finite is Zariski-dense in the character variety Y(g,n,r).

Background

This conjecture is presented as a proposed partial answer to the problem of classifying finite orbits of the Mod_{g,n}-action on character varieties Y(g,n,r). It stems from arithmetic-geometric perspectives that suggest local systems of geometric origin should be plentiful in moduli, and it would imply that finite orbits occur densely in the space of representations—even though later results in the paper show this particular conjecture is false in general.