Path-connectivity of the set of uniquely ergodic and cobounded foliations

We show that if S is a closed surface of genus \(g\ge 5\) $g\ge 5$ or a surface of genus \(g\ge 2\) $g\ge 2$ with at least \(p\ge 1\) $p\ge 1$ marked points, then the set of uniquely ergodic foliations and the set of cobounded foliations is path-connected and locally path-connected.