diff --git a/include/haproxy/connection-t.h b/include/haproxy/connection-t.h
index 531405607..7cac07a69 100644
--- a/include/haproxy/connection-t.h
+++ b/include/haproxy/connection-t.h
@@ -424,6 +424,14 @@ struct mux_ops {
 	char name[8];                                 /* mux layer name, zero-terminated */
 };
 
+/* list of frontend connections. Used to call mux wake operation on soft-stop
+ * to close idling connections.
+ */
+struct mux_stopping_data {
+	struct list list; /* list of registered frontend connections */
+	struct task *task; /* task woken up on soft-stop */
+};
+
 /* data_cb describes the data layer's recv and send callbacks which are called
  * when I/O activity was detected after the transport layer is ready. These
  * callbacks are supposed to make use of the xprt_ops above to exchange data
@@ -528,6 +536,7 @@ struct connection {
 	struct mt_list toremove_list; /* list for connection to clean up */
 	union {
 		struct list session_list;  /* used by backend conns, list of attached connections to a session */
+		struct list stopping_list; /* used by frontend conns, attach point in mux stopping list */
 	};
 	union conn_handle handle;     /* connection handle at the socket layer */
 	const struct netns_entry *proxy_netns;
diff --git a/include/haproxy/connection.h b/include/haproxy/connection.h
index 296da89e5..b0b855e2c 100644
--- a/include/haproxy/connection.h
+++ b/include/haproxy/connection.h
@@ -44,6 +44,7 @@ extern struct pool_head *pool_head_sockaddr;
 extern struct pool_head *pool_head_authority;
 extern struct xprt_ops *registered_xprt[XPRT_ENTRIES];
 extern struct mux_proto_list mux_proto_list;
+extern struct mux_stopping_data mux_stopping_data[MAX_THREADS];
 
 #define IS_HTX_CONN(conn) ((conn)->mux && ((conn)->mux->flags & MX_FL_HTX))
 #define IS_HTX_CS(cs)     (IS_HTX_CONN((cs)->conn))
@@ -373,6 +374,8 @@ static inline void conn_init(struct connection *conn, void *target)
 	MT_LIST_INIT(&conn->toremove_list);
 	if (conn_is_back(conn))
 		LIST_INIT(&conn->session_list);
+	else
+		LIST_INIT(&conn->stopping_list);
 	conn->subs = NULL;
 	conn->src = NULL;
 	conn->dst = NULL;
diff --git a/include/haproxy/global.h b/include/haproxy/global.h
index 9d8f7896e..4231ba829 100644
--- a/include/haproxy/global.h
+++ b/include/haproxy/global.h
@@ -57,6 +57,8 @@ extern int atexit_flag;
 extern unsigned char boot_seed[20];  // per-boot random seed (160 bits initially)
 extern THREAD_LOCAL struct buffer trash;
 
+extern struct task *stopping_task[MAX_PROCS];
+
 struct proxy;
 struct server;
 int main(int argc, char **argv);
diff --git a/src/connection.c b/src/connection.c
index c0ba8155f..67b212452 100644
--- a/src/connection.c
+++ b/src/connection.c
@@ -41,6 +41,8 @@ struct mux_proto_list mux_proto_list = {
         .list = LIST_HEAD_INIT(mux_proto_list.list)
 };
 
+struct mux_stopping_data mux_stopping_data[MAX_THREADS];
+
 /* disables sending of proxy-protocol-v2's LOCAL command */
 static int pp2_never_send_local;
 
diff --git a/src/haproxy.c b/src/haproxy.c
index 5d3e735e2..f149fc5cb 100644
--- a/src/haproxy.c
+++ b/src/haproxy.c
@@ -2370,11 +2370,31 @@ __attribute__((noreturn)) void deinit_and_exit(int status)
 	exit(status);
 }
 
+/* Handler of the task of mux_stopping_data.
+ * Called on soft-stop.
+ */
+struct task *mux_stopping_process(struct task *t, void *ctx, unsigned int state)
+{
+	struct connection *conn, *back;
+
+	list_for_each_entry_safe(conn, back, &mux_stopping_data[tid].list, stopping_list) {
+		if (conn->mux && conn->mux->wake)
+			conn->mux->wake(conn);
+	}
+
+	return t;
+}
+
 /* Runs the polling loop */
 void run_poll_loop()
 {
 	int next, wake;
 
+	/* allocates the thread bound mux_stopping_data task */
+	mux_stopping_data[tid].task = task_new(tid_bit);
+	mux_stopping_data[tid].task->process = mux_stopping_process;
+	LIST_INIT(&mux_stopping_data[tid].list);
+
 	tv_update_date(0,1);
 	while (1) {
 		wake_expired_tasks();
@@ -2411,6 +2431,12 @@ void run_poll_loop()
 			int i;
 
 			if (stopping) {
+				/* stop muxes before acknowleding stopping */
+				if (!(stopping_thread_mask & tid_bit)) {
+					task_wakeup(mux_stopping_data[tid].task, TASK_WOKEN_OTHER);
+					wake = 1;
+				}
+
 				if (_HA_ATOMIC_OR_FETCH(&stopping_thread_mask, tid_bit) == tid_bit) {
 					/* notify all threads that stopping was just set */
 					for (i = 0; i < global.nbthread; i++)
@@ -2438,6 +2464,8 @@ void run_poll_loop()
 
 		activity[tid].loops++;
 	}
+
+	task_destroy(mux_stopping_data[tid].task);
 }
 
 static void *run_thread_poll_loop(void *data)
diff --git a/src/mux_h1.c b/src/mux_h1.c
index 940b7540c..62bd4771d 100644
--- a/src/mux_h1.c
+++ b/src/mux_h1.c
@@ -816,6 +816,9 @@ static int h1_init(struct connection *conn, struct proxy *proxy, struct session
 		h1c->shut_timeout = h1c->timeout = proxy->timeout.client;
 		if (tick_isset(proxy->timeout.clientfin))
 			h1c->shut_timeout = proxy->timeout.clientfin;
+
+		LIST_APPEND(&mux_stopping_data[tid].list,
+		            &h1c->conn->stopping_list);
 	}
 	if (tick_isset(h1c->timeout)) {
 		t = task_new(tid_bit);
@@ -936,6 +939,9 @@ static void h1_release(struct h1c *h1c)
 	}
 
 	if (conn) {
+		if (!conn_is_back(conn))
+			LIST_DEL_INIT(&conn->stopping_list);
+
 		conn->mux = NULL;
 		conn->ctx = NULL;
 		TRACE_DEVEL("freeing conn", H1_EV_H1C_END, conn);
diff --git a/src/mux_h2.c b/src/mux_h2.c
index ebae4382c..1374cd7e0 100644
--- a/src/mux_h2.c
+++ b/src/mux_h2.c
@@ -955,6 +955,15 @@ static int h2_init(struct connection *conn, struct proxy *prx, struct session *s
 	h2c->wait_event.tasklet->process = h2_io_cb;
 	h2c->wait_event.tasklet->context = h2c;
 	h2c->wait_event.events = 0;
+	if (!conn_is_back(conn)) {
+		/* Connection might already be in the stopping_list if subject
+		 * to h1->h2 upgrade.
+		 */
+		if (!LIST_INLIST(&conn->stopping_list)) {
+			LIST_APPEND(&mux_stopping_data[tid].list,
+			            &conn->stopping_list);
+		}
+	}
 
 	h2c->ddht = hpack_dht_alloc();
 	if (!h2c->ddht)
@@ -1097,6 +1106,9 @@ static void h2_release(struct h2c *h2c)
 	}
 
 	if (conn) {
+		if (!conn_is_back(conn))
+			LIST_DEL_INIT(&conn->stopping_list);
+
 		conn->mux = NULL;
 		conn->ctx = NULL;
 		TRACE_DEVEL("freeing conn", H2_EV_H2C_END, conn);