In my scenario at time N, the connection is idle -- both sides have received all data the other has sent, all requests have received a response.

If the server half-closes (through shutdown) and sends a FIN, simultaneously with the client sending a new request; that enables the server to read the request, but not respond to it, so I don't see how that is helpful?

The problem from the client side is it's sent a request, and seemingly in response the socket is closed. That could indicate the server crashed on the request, or the server closed the socket because it was idle. If you have a request that you know or suspect shouldn't be made more than once, you shouldn't retry it on a new connection. Assuming the tcp packets from the server, you can actually take a good guess at causality, because the ACK number and TCP Timestamp indicate if the server saw your last transmission, but that information isn't exposed through normal the normal socket API; you could maybe guess based on round trip time too, but it is nicer in http/2 (or other protocols), where there is an explicit close message.