diff --git a/internal/rproxy/limits_test.go b/internal/rproxy/limits_test.go
index 2af3b1f..4247692 100644
--- a/internal/rproxy/limits_test.go
+++ b/internal/rproxy/limits_test.go
@@ -12,7 +12,6 @@ import (
 	"os"
 	"strconv"
 	"strings"
-	"sync"
 	"sync/atomic"
 	"testing"
 	"time"
@@ -918,49 +917,48 @@ func TestAVisitorThatStopsALittleShortHasTheConnectionClosedWhenTheAnswerIsCutOf
 ) {
 	t.Parallel()
 
-	const limit = 300 * time.Millisecond
+	// A busy host passes on a part sent at once well within this limit.
+	const limit = 2 * time.Second
 
-	// With two parts, rproxy fails to send the first and then waits for the
-	// second, which the app sends only once rproxy has ended the request to it.
+	// The app sends the parts in atOnce with its status and headers. It sends the
+	// parts in late only once rproxy has ended the request to it, which rproxy
+	// does when it cuts off the answer, so that those never reach the visitor.
 	for _, tc := range []struct {
-		name  string
-		parts []string
+		name         string
+		atOnce, late []string
 	}{
-		{"with the body in one part", []string{greeting}},
-		{"with the body in two parts", []string{greeting[:5], greeting[5:]}},
+		{"with the body in one part", nil, []string{greeting}},
+		{"with the body in two parts", nil, []string{greeting[:5], greeting[5:]}},
+		{
+			"with the first part of the body at once",
+			[]string{greeting[:5]}, []string{greeting[5:]},
+		},
 	} {
 		t.Run(tc.name, func(t *testing.T) {
 			t.Parallel()
 
-			headersRead := make(chan struct{})
-			// letAppSendBody also runs when the test ends, as the app's server waits
-			// for the app.
-			letAppSendBody := sync.OnceFunc(func() { close(headersRead) })
 			standIn := startWAF(t, allow)
-			// The app sends its body once the visitor has the status and headers and
-			// the response time limit has run out since, so that rproxy cannot send
-			// it. After each part, it reads the request's body, which ends when rproxy
-			// ends the request to it.
 			app := startApp(t, func(w http.ResponseWriter, r *http.Request) {
 				rc := http.NewResponseController(w)
 				_ = rc.EnableFullDuplex()
 
 				w.Header().Set(contentLengthHeader, strconv.Itoa(len(greeting)))
 
+				for _, part := range tc.atOnce {
+					_, _ = io.WriteString(w, part)
+				}
+
 				_ = rc.Flush()
 
-				<-headersRead
-				time.Sleep(limit)
+				// The request's body ends when rproxy ends the request to the app.
+				_, _ = io.Copy(io.Discard, r.Body)
 
-				for _, part := range tc.parts {
+				for _, part := range tc.late {
 					_, _ = io.WriteString(w, part)
 					_ = rc.Flush()
-					_, _ = io.Copy(io.Discard, r.Body)
 				}
 			})
 
-			t.Cleanup(letAppSendBody)
-
 			// The request time limit is far longer than dial's deadline.
 			addr, _ := startRProxy(t, standIn.URL, app.URL, map[string]string{
 				maxRequestTime:  "120000",
@@ -975,15 +973,14 @@ func TestAVisitorThatStopsALittleShortHasTheConnectionClosedWhenTheAnswerIsCutOf
 				t.Fatalf("read the answer: %v", err)
 			}
 
-			letAppSendBody()
-
 			// If the connection stays open, dial's deadline ends this read with a
 			// timeout.
 			body, err := io.ReadAll(res.Body)
 			_ = res.Body.Close()
 
-			if len(body) != 0 || !errors.Is(err, io.ErrUnexpectedEOF) {
-				t.Errorf("read %q (%v), want the connection closed", body, err)
+			want := strings.Join(tc.atOnce, "")
+			if string(body) != want || !errors.Is(err, io.ErrUnexpectedEOF) {
+				t.Errorf("read %q (%v), want %q and the connection closed", body, err, want)
 			}
 		})
 	}