diff --git a/src/test/ssl/t/004_sni.pl b/src/test/ssl/t/004_sni.pl
index 0542c59ebcb..e183a953ede 100644
--- a/src/test/ssl/t/004_sni.pl
+++ b/src/test/ssl/t/004_sni.pl
@@ -57,6 +57,32 @@ $node->connect_ok(
 	"$connstr sslrootcert=ssl/root+server_ca.crt sslmode=require",
 	"connect with correct server CA cert file sslmode=require");
 
+# This is added only for comparison with the same test case below.
+$node->connect_fails(
+	"$connstr sslrootcert=ssl/root_ca.crt sslmode=verify-ca",
+	"connect fails with fallback hostname, without intermediate",
+	expected_stderr => qr/certificate verify failed/);
+
+# example.org serves the server cert and its intermediate CA.
+$node->append_conf('pg_hosts.conf',
+	"example.org server-cn-only+server_ca.crt server-cn-only.key root_ca.crt"
+);
+$node->reload;
+
+$node->connect_ok(
+	"$connstr host=example.org sslrootcert=ssl/root_ca.crt sslmode=verify-ca",
+	"connect with configured hostname, serving intermediate server CA");
+
+# Why does this test fail?
+$node->connect_fails(
+	"$connstr sslrootcert=ssl/root_ca.crt sslmode=verify-ca",
+	"connect still fails with fallback hostname, without intermediate",
+	expected_stderr => qr/certificate verify failed/);
+
+$node->connect_ok(
+	"$connstr host=localhost sslrootcert=ssl/root+server_ca.crt sslmode=verify-ca",
+	"connect with fallback hostname, intermediate included");
+
 ok(unlink($node->data_dir . '/pg_hosts.conf'));
 $node->append_conf('pg_hosts.conf',
 	"localhost server.crt server.key root.crt");