diff --git a/pageserver/client/src/page_service.rs b/pageserver/client/src/page_service.rs
index 231461267a..ff542670f1 100644
--- a/pageserver/client/src/page_service.rs
+++ b/pageserver/client/src/page_service.rs
@@ -108,9 +108,32 @@ pub struct RelTagBlockNo {
 }
 
 impl PagestreamClient {
-    pub async fn shutdown(mut self) {
-        let _ = self.cancel_on_client_drop.take();
-        self.conn_task.await.unwrap();
+    pub async fn shutdown(self) {
+        let Self {
+            copy_both,
+            cancel_on_client_drop: cancel_conn_task,
+            conn_task,
+        } = self;
+        // The `copy_both` contains internal channel sender, the receiver of which is polled by `conn_task`.
+        // When `conn_task` observes the sender has been dropped, it sends a `FeMessage::CopyFail` into the connection.
+        // (see https://github.com/neondatabase/rust-postgres/blob/2005bf79573b8add5cf205b52a2b208e356cc8b0/tokio-postgres/src/copy_both.rs#L56).
+        //
+        // If we drop(copy_both) first, but then immediately drop the `cancel_on_client_drop`,
+        // the CopyFail mesage only makes it to the socket sometimes (i.e., it's a race).
+        //
+        // Further, the pageserver makes a lot of noise when it receives CopyFail.
+        // Computes don't send it in practice, they just hard-close the connection.
+        //
+        // So, let's behave like the computes and suppress the CopyFail as follows:
+        // kill the socket first, then drop copy_both.
+        //
+        // See also: https://www.postgresql.org/docs/current/protocol-flow.html#PROTOCOL-COPY
+        //
+        // NB: page_service doesn't have a use case to exit the `pagestream` mode currently.
+        // => https://github.com/neondatabase/neon/issues/6390
+        let _ = cancel_conn_task.unwrap();
+        conn_task.await.unwrap();
+        drop(copy_both);
     }
 
     pub async fn getpage(
diff --git a/pageserver/pagebench/src/cmd/getpage_latest_lsn.rs b/pageserver/pagebench/src/cmd/getpage_latest_lsn.rs
index d8957ddd6b..98f1852acd 100644
--- a/pageserver/pagebench/src/cmd/getpage_latest_lsn.rs
+++ b/pageserver/pagebench/src/cmd/getpage_latest_lsn.rs
@@ -404,23 +404,27 @@ async fn client(
         .await
         .unwrap();
 
-    start_work_barrier.wait().await;
-
-    while let Some(req) =
-        tokio::select! { work = work.recv() => { work } , _ = cancel.cancelled() => { return; } }
-    {
-        let start = Instant::now();
-
-        let res = tokio::select! {
-            res = client.getpage(req) => { res },
-            _ = cancel.cancelled() => { return; }
-        };
-        res.with_context(|| format!("getpage for {timeline}"))
-            .unwrap();
-        let elapsed = start.elapsed();
-        live_stats.inc();
-        STATS.with(|stats| {
-            stats.borrow().lock().unwrap().observe(elapsed).unwrap();
-        });
+    let do_requests = async {
+        start_work_barrier.wait().await;
+        while let Some(req) = work.recv().await {
+            let start = Instant::now();
+            client
+                .getpage(req)
+                .await
+                .with_context(|| format!("getpage for {timeline}"))
+                .unwrap();
+            let elapsed = start.elapsed();
+            live_stats.inc();
+            STATS.with(|stats| {
+                stats.borrow().lock().unwrap().observe(elapsed).unwrap();
+            });
+        }
+    };
+    tokio::select! {
+        res = do_requests => { res },
+        _ = cancel.cancelled() => {
+            client.shutdown().await;
+            return;
+        }
     }
 }