A couple of points to note:
The analysis in the recent posts by Borntoski and jamt is generally correct.
However there is no difference between extending on the flat bottom and the lower part of the curved side.
When we stand up on a curved surface we do NOT actually speed up. The angular rate (theta dot = d(theta)/dt) goes up, but the speed is (radius)*(theta dot) and that stays the same.
So far we have not done any front foot/back foot analysis. The math had been for a single point of support in line with the center of mass. The front-back effects are a very interesting question. My hunch is that it provides many additional ways to pump, but that is only a hunch.
I think that if you accelerate forward by pressing more on the tails, and then bring the skis back under you unweighted, you get a pumping cycle with a net energy input.