secrec Keeping a "persistent connection" doesn't mean that you are continually doing something. It means that you've instructed the ROUTERS between you and the target server on how to deliver messages to you from that server.
That's not how the Internet works. If one is at home and one's home router uses NAT, the NAT function's state table must record the connection. But the rest of the routers that make up the Internet are not required to retain any information about the connection, and they essentially never do. "Because routing is dynamic, meaning every packet is treated independently, and because the network maintains no state based on the path of prior packets, different packets may be routed to the same destination via different paths, resulting in out-of-order delivery to the receiver." (source).