Wrinkly Rug.

I think if it if it fully lined the lining is the problem.
we have a large rug on a carpet and I bought an antislip liner thingy from John Lewis, it was about £60 and worth every penny .You cut it to size and it goes under the whole rug. I agree though that it may be your carpet that is the problem.
Good luck especially if you love both carpet and rug.

a rug on a carpet is an accident waiting to happen

I give up. I'll just try to live with it and not fall over the kink in the middle.

Is it possible that where the wrinkle is, the top part of the rug has become separate from the material or hessian on the underneath section of the rug?

We had this problem, too! In the end, we used grippers plus double-sided carpet tape. Lots of it! So far, so good .....

Looks like rug need to be stretched and weighted down at the four corners.

I think you are right re the thickness of the pile underneath your rug.
Try turning the rug over for a couple of days that might help with wrinkle in middle

Long shot
but you never know

I had this problem and could not prevent waves in the rug, lost patience and gave it to DD who never had a problem. I think the difference was my long pile, very expensive wool carpet and she had thin mixed material covering.

Done that roses, it didn't work! I'm at a loss. I've trawled the internet looking for ideas but found none! We did have a heavy coffee table on top of it, but the rug just wrinkled up in the same manner.

Try taking up the anti slip mat underneath, then relaying it and walking all over it.
Then lay your rug on it and do the same.

Strange! This often happens when heavy furniture sits atop a rug, but not when placed otherwise.? Looks a bit dangerous too.What I think, after a ponder, is that the carpet is heavyweight and the rug lightweight.As you walk across it it moves into the middle, has no grip at all.All down to the long pile carpet!