diff --git a/t/t5515-fetch-merge-logic.sh b/t/t5515-fetch-merge-logic.sh
index 36b0dbc01c..e55d8474ef 100755
--- a/t/t5515-fetch-merge-logic.sh
+++ b/t/t5515-fetch-merge-logic.sh
@@ -6,6 +6,10 @@
 
 test_description='Merge logic in fetch'
 
+# NEEDSWORK: If the overspecification of the expected result is reduced, we
+# might be able to run this test in all protocol versions.
+GIT_TEST_PROTOCOL_VERSION=
+
 . ./test-lib.sh
 
 LF='
diff --git a/t/t5539-fetch-http-shallow.sh b/t/t5539-fetch-http-shallow.sh
index 5fbf67c446..cdb687b93a 100755
--- a/t/t5539-fetch-http-shallow.sh
+++ b/t/t5539-fetch-http-shallow.sh
@@ -67,7 +67,10 @@ test_expect_success 'no shallow lines after receiving ACK ready' '
 		cd clone &&
 		git checkout --orphan newnew &&
 		test_commit new-too &&
-		GIT_TRACE_PACKET="$TRASH_DIRECTORY/trace" git fetch --depth=2 &&
+		# NEEDSWORK: If the overspecification of the expected result is reduced, we
+		# might be able to run this test in all protocol versions.
+		GIT_TRACE_PACKET="$TRASH_DIRECTORY/trace" GIT_TEST_PROTOCOL_VERSION= \
+			git fetch --depth=2 &&
 		grep "fetch-pack< ACK .* ready" ../trace &&
 		! grep "fetch-pack> done" ../trace
 	)
diff --git a/t/t5541-http-push-smart.sh b/t/t5541-http-push-smart.sh
index 5475afc052..0e3055ab98 100755
--- a/t/t5541-http-push-smart.sh
+++ b/t/t5541-http-push-smart.sh
@@ -47,7 +47,12 @@ test_expect_success 'no empty path components' '
 	cd "$ROOT_PATH" &&
 	git clone $HTTPD_URL/smart/test_repo.git/ test_repo_clone &&
 
-	check_access_log exp
+	# NEEDSWORK: If the overspecification of the expected result is reduced, we
+	# might be able to run this test in all protocol versions.
+	if test -z "$GIT_TEST_PROTOCOL_VERSION"
+	then
+		check_access_log exp
+	fi
 '
 
 test_expect_success 'clone remote repository' '
@@ -128,7 +133,12 @@ GET  /smart/test_repo.git/info/refs?service=git-receive-pack HTTP/1.1 200
 POST /smart/test_repo.git/git-receive-pack HTTP/1.1 200
 EOF
 test_expect_success 'used receive-pack service' '
-	check_access_log exp
+	# NEEDSWORK: If the overspecification of the expected result is reduced, we
+	# might be able to run this test in all protocol versions.
+	if test -z "$GIT_TEST_PROTOCOL_VERSION"
+	then
+		check_access_log exp
+	fi
 '
 
 test_http_push_nonff "$HTTPD_DOCUMENT_ROOT_PATH"/test_repo.git \
diff --git a/t/t5551-http-fetch-smart.sh b/t/t5551-http-fetch-smart.sh
index d2661eb338..e8ed185b33 100755
--- a/t/t5551-http-fetch-smart.sh
+++ b/t/t5551-http-fetch-smart.sh
@@ -82,12 +82,18 @@ test_expect_success 'clone http repository' '
 		/^< Content-Length: /d
 		/^< Transfer-Encoding: /d
 	" >actual &&
-	sed -e "s/^> Accept-Encoding: .*/> Accept-Encoding: ENCODINGS/" \
-			actual >actual.smudged &&
-	test_cmp exp actual.smudged &&
 
-	grep "Accept-Encoding:.*gzip" actual >actual.gzip &&
-	test_line_count = 2 actual.gzip
+	# NEEDSWORK: If the overspecification of the expected result is reduced, we
+	# might be able to run this test in all protocol versions.
+	if test -z "$GIT_TEST_PROTOCOL_VERSION"
+	then
+		sed -e "s/^> Accept-Encoding: .*/> Accept-Encoding: ENCODINGS/" \
+				actual >actual.smudged &&
+		test_cmp exp actual.smudged &&
+
+		grep "Accept-Encoding:.*gzip" actual >actual.gzip &&
+		test_line_count = 2 actual.gzip
+	fi
 '
 
 test_expect_success 'fetch changes via http' '
@@ -105,7 +111,13 @@ test_expect_success 'used upload-pack service' '
 	GET  /smart/repo.git/info/refs?service=git-upload-pack HTTP/1.1 200
 	POST /smart/repo.git/git-upload-pack HTTP/1.1 200
 	EOF
-	check_access_log exp
+
+	# NEEDSWORK: If the overspecification of the expected result is reduced, we
+	# might be able to run this test in all protocol versions.
+	if test -z "$GIT_TEST_PROTOCOL_VERSION"
+	then
+		check_access_log exp
+	fi
 '
 
 test_expect_success 'follow redirects (301)' '
@@ -217,8 +229,14 @@ test_expect_success 'cookies stored in http.cookiefile when http.savecookies set
 	git config http.cookiefile cookies.txt &&
 	git config http.savecookies true &&
 	git ls-remote $HTTPD_URL/smart_cookies/repo.git master &&
-	tail -3 cookies.txt | sort >cookies_tail.txt &&
-	test_cmp expect_cookies.txt cookies_tail.txt
+
+	# NEEDSWORK: If the overspecification of the expected result is reduced, we
+	# might be able to run this test in all protocol versions.
+	if test -z "$GIT_TEST_PROTOCOL_VERSION"
+	then
+		tail -3 cookies.txt | sort >cookies_tail.txt &&
+		test_cmp expect_cookies.txt cookies_tail.txt
+	fi
 '
 
 test_expect_success 'transfer.hiderefs works over smart-http' '
