How can I register a session supervised prepared query for another data center?

I use session supervised prepared queries to find services within the same data center as my agent. Now I want to be able to execute session supervised queries for other specified data centers but can’t seem to figure out how to make it work.

For the own data center it is straight forward, create a session, create a query with the session ID from the session create and execute when required:

curl -X PUT -d '{"TTL":"10s"}'
curl -X POST -d '{"Session":"a1a8bf4d-9e3d-2236-bec8-1cbfb8c8bdd0","Service":{"Service":"TheService","OnlyPassing":true,"Tags":["sometag"]}}'

I assumed that I could simply add the URL query parameter ?dc=<name> to these commands, which works as long as I specify my own data center, but I fail immediately on session create with error: rpc error making call: Missing node registration if a specify another data center in the WAN. I have also tried other options like registering the session and query without the dc parameter and only use it in the execute for instance. Long shot, got unkown query or similar, not unexpected.

Is this a supported use-case and if so, how can I accomplish this?

Any help much appreciated.