What is the advantage of a reverse arch suspension fork?
Looks like nobody has taken the geometry into account yet...

In the image below I have drawn the situation with highly exaggerated proportions. 



All not-lowest-end suspension forks (at least the ones I know of) have a rake that is created by shifting the axle in front of the lower tubes. Let's take a wheel of radius r (solid blue circle) and add some clearance for the tire (dashed blue circle). Above that, the upper edge of the lower tubes shall be of equal height with respect to the upper tube's axis.

Here we can already see, that the wheel passes the front side of the lowers at a "higher" position than its back side. As the arches (green rectangles) have to span from one lower tube to the other over the clearance radius, we can see that the arch in the front side (the right one in the image) has to reach higher (higher green rectangle) above the lower tube's body than the one on the back side.

Technically speaking this means, that to build an arch with the same stiffness, less material (i.e. less weight) is needed on a reversed arch or at the same weight, the reversed arch can be built more massive and therefore stiffer. Unfortunately I don't have any measurement values about how much influence that really has. And obviously there are (or at least were, cf. heltonbike's answer) other drawbacks to be taken into account.