Puppet with a dog on a leash

Hello there! This is my first post here, and it is to ask for help.
I would like to make a character that holds a dog on a leash.
I don't want to make the dog move I just want the movement of the leash to be a bit realistic.
I added in the swap set a hand with a leash and the dog.

I added some fixed handel on the dog
Then, I get a leash movement that I like but when I move my head the dog moves at the same time, which does not suit me (screen.mov file).
Is it possible to make the dog stay put?

Thank you in advance for your help

Kind regards

Thierry