diff --git a/Dockerfile b/Dockerfile
index 30aace9..79d3ef4 100644
--- a/Dockerfile
+++ b/Dockerfile
@@ -29,7 +29,10 @@ RUN go mod download
 
 COPY . .
 
-RUN go test -count=1 -timeout 90s -race -cover ./... || \
+# Go's build cache is kept on a tmpfs, out of the image: nothing uses it
+# after this step, and writing it into the image takes seconds.
+RUN --mount=type=tmpfs,target=/root/.cache/go-build \
+    go test -count=1 -timeout 90s -race -cover ./... || \
     { echo "--- Rerunning with -v for details ---"; \
       go test -count=1 -timeout 90s -race -v ./...; exit 1; }
 
diff --git a/internal/proxy/passthrough_test.go b/internal/proxy/passthrough_test.go
index 8f644bb..c278a60 100644
--- a/internal/proxy/passthrough_test.go
+++ b/internal/proxy/passthrough_test.go
@@ -240,15 +240,17 @@ func TestUpgradedConnectionOutlastsTheTimeouts(t *testing.T) {
 		t.Fatalf("read the answer to the upgrade: %v", err)
 	}
 
+	answered := time.Now()
 	_ = res.Body.Close()
 
 	if res.StatusCode != http.StatusSwitchingProtocols {
 		t.Fatalf("status %d, want %d", res.StatusCode, http.StatusSwitchingProtocols)
 	}
 
-	// Every timeout started before the upgrade was answered: wait past
-	// them all, then use the connection.
-	time.Sleep(3 * shortTimeout / 2)
+	// Every timeout started before the upgrade was answered, the response
+	// timeouts last, at the end of the request: wait until just past
+	// shortTimeout after the answer was read, then use the connection.
+	time.Sleep(time.Until(answered.Add(shortTimeout + 100*time.Millisecond)))
 	send(t, conn, "still here\n")
 
 	echoed, err := reader.ReadString('\n')
diff --git a/internal/proxy/proxy_test.go b/internal/proxy/proxy_test.go
index 0e7dea7..24b33f4 100644
--- a/internal/proxy/proxy_test.go
+++ b/internal/proxy/proxy_test.go
@@ -23,9 +23,9 @@ const (
 	// shortTimeout is what a test sets a timeout to, to see it run out.
 	// It starts before the test has set up its case, such as an upgrade
 	// or the app's buffers filling, so it is as long as the hold-up of the
-	// test process that wantTimedOut allows, waitLimit/2: a shorter one
-	// can run out first on a busy host.
-	shortTimeout = 5 * time.Second
+	// test process that wantTimedOut allows: a shorter one can run out
+	// first on a busy host.
+	shortTimeout = waitLimit / 2
 	// shortTimeoutSetting is shortTimeout as a setting's value.
 	shortTimeoutSetting = "5s"
 	// longTimeoutSetting is a timeout that does not run out in a test.