As is mentioned in a few other posts, the hypersphere does not constitute an inertial frame, which means you can't blindly apply SR to it.
As Tom Mattson suggested, one should work out the case of two travellers who, in flat space, travel away from each other, turn around, and return. It seems to me that this shows the dynamics of the issue more clearly. They hypersphere scenario is similar, but has a small acceleration over the whole trip, and it makes it harder to see what's going on. Though once you've solved the simpler case, it will make the hypersphere case a little easier to understand. Walk before you run, and all that.