I did a little research and could not verify that SD has such a peculiar law. Turns out that they don't

I finally talked to a guy at F&G HQ.

In SD you can pick up sheds on private land and federal land like BLM and US Forest Service land. No Problem.

You cannot collect shed antlers or any other natural artifacts on South Dakota State Park or other State lands.

You cannot pick up antlers still attached to a skull or skullplate anywhere. The hunting regulations are clear on that point as well.

So anyone found any dandy sheds yet?