I see from the ssh_config man page that you can share connections to host@user using ControlPath and ControlMaster.
Is there any way to do this opportunistically for each combination of local user, remote user, and remote host? So that the first connection to testuser@remotelaptop is shared when you ssh into it the second time and the first is still running?
(I use ssh keys. I'm not sure if that would make it easier or harder).