Sticky Sessions / Session affinity cookies for multiple services

I am setting a sticky lb cookie like this:

           name: istiolb

If I have multiple services / Destination rules, does the cookie need to be unique for each service, or can I share the same load balancer cookie name across services?

Following up on my question…

I see that the istio sets the path for cookie, as well as the host. So in theory if the applications have different paths, they could share the same cookie. Do I have that right?