As of #96363 a shutting-down Elasticsearch node will close HTTP connections as soon as they become idle. However if the client is using HTTP pipelining then it may be able to keep the connection open indefinitely without knowing that the node is trying to shut down.

We should set the Connection: close header on responses sent after we start to shut down so that the client can react by stopping its use of the connection to the shutting-down node, even if it is pipelining requests.