In the paper "The Nielsen realization problem" (here), Kerckhoff proved that the length function on the Teichmüller for closed surface is convex. In his paper "Geodesic length functions and the Nielsen problem" here, Wolpert gave an alternative proof which includes the convexity of length functions for surfaces with punctures.

Is the length function is also convex on the Teichmüller spaces of surfaces with boundary? If so does it follow from Wolpert's result?

In the paper "The Nielsen realization problem" (here), Kerckhoff proved that the length function on the Teichmüller for closed surface is convex. In his paper "Geodesic length functions and the Nielsen problem" here, Wolpert gave an alternative proof which includes the convexity of length functions for surfaces with punctures.

Kerckhoff's proof shows convexity along earthquake paths, on the other hand Wolpert proved the convexity along Weil-Petersson Geodesics. So the question is

1) Is the length function convex along earthquake paths on the Teichmüller spaces of surfaces with boundary and punctures?

2) Is the length function convex along Weil-Petersson geodesics on the Teichmüller spaces of surfaces with boundary?