A quick tweet to ServerGrove triggered me to receive this link from them with a way to possibly prevent this. I took the server-side version, and added to my /etc/ssh/sshd_config:
I restarted my ssh daemon and connected again. I then went to pick up my son with the connection still open, and when I came back... no stalling anymore. I could still use the connection! So this seems to have solved this problem for me. Hopefully, it will for you as well. Yay for ServerGrove!