diff --git a/src/mux_h1.c b/src/mux_h1.c
index 53660f3..ecdb381 100644
--- a/src/mux_h1.c
+++ b/src/mux_h1.c
@@ -591,6 +591,8 @@
 /* Refresh the h1c task timeout if necessary */
 static void h1_refresh_timeout(struct h1c *h1c)
 {
+	int is_idle_conn = 0;
+
 	if (h1c->task) {
 		if (!(h1c->flags & H1C_F_ST_ALIVE) || (h1c->flags & H1C_F_ST_SHUTDOWN)) {
 			/* half-closed or dead connections : switch to clientfin/serverfin
@@ -599,6 +601,7 @@
 			 */
 			h1c->task->expire = tick_add(now_ms, h1c->shut_timeout);
 			TRACE_DEVEL("refreshing connection's timeout (dead or half-closed)", H1_EV_H1C_SEND|H1_EV_H1C_RECV, h1c->conn);
+			is_idle_conn = 1;
 		}
 		else if (b_data(&h1c->obuf)) {
 			/* connection with pending outgoing data, need a timeout (server or client). */
@@ -609,6 +612,10 @@
 			/* front connections waiting for a fully usable stream need a timeout. */
 			h1c->task->expire = tick_add(now_ms, h1c->timeout);
 			TRACE_DEVEL("refreshing connection's timeout (alive front h1c but not ready)", H1_EV_H1C_SEND|H1_EV_H1C_RECV, h1c->conn);
+			/* A frontend connection not yet ready could be treated the same way as an idle
+			 * one in case of soft-close.
+			 */
+			is_idle_conn = 1;
 		}
 		else  {
 			/* alive back connections of front connections with a conn-stream attached */
@@ -618,6 +625,36 @@
 
 		/* Finally set the idle expiration date if shorter */
 		h1c->task->expire = tick_first(h1c->task->expire, h1c->idle_exp);
+
+		if ((h1c->px->flags & (PR_FL_DISABLED|PR_FL_STOPPED)) &&
+		     is_idle_conn && tick_isset(global.close_spread_end)) {
+			/* If a soft-stop is in progress and a close-spread-time
+			 * is set, we want to spread idle connection closing roughly
+			 * evenly across the defined window. This should only
+			 * act on idle frontend connections.
+			 * If the window end is already in the past, we wake the
+			 * timeout task up immediately so that it can be closed.
+			 */
+			int remaining_window = tick_remain(now_ms, global.close_spread_end);
+			if (remaining_window) {
+				/* We don't need to reset the expire if it would
+				 * already happen before the close window end.
+				 */
+				if (tick_is_le(global.close_spread_end, h1c->task->expire)) {
+					/* Set an expire value shorter than the current value
+					 * because the close spread window end comes earlier.
+					 */
+					h1c->task->expire = tick_add(now_ms, statistical_prng_range(remaining_window));
+					TRACE_DEVEL("connection timeout set to value before close-spread window end", H1_EV_H1C_SEND|H1_EV_H1C_RECV, h1c->conn);
+				}
+			}
+			else {
+				/* We are past the soft close window end, wake the timeout
+				 * task up immediately.
+				 */
+				task_wakeup(h1c->task, TASK_WOKEN_TIMER);
+			}
+		}
 		TRACE_DEVEL("new expiration date", H1_EV_H1C_SEND|H1_EV_H1C_RECV, h1c->conn, 0, 0, (size_t[]){h1c->task->expire});
 		task_queue(h1c->task);
 	}
@@ -3011,8 +3048,25 @@
 	if (!(h1c->flags & H1C_F_IS_BACK)) {
 		if (unlikely(h1c->px->flags & (PR_FL_DISABLED|PR_FL_STOPPED))) {
 			if (!(h1c->px->options & PR_O_IDLE_CLOSE_RESP) &&
-				h1c->flags & H1C_F_WAIT_NEXT_REQ)
-				goto release;
+				h1c->flags & H1C_F_WAIT_NEXT_REQ) {
+
+				int send_close = 1;
+				/* If a close-spread-time option is set, we want to avoid
+				 * closing all the active HTTP2 connections at once so we add a
+				 * random factor that will spread the closing.
+				 */
+				if (tick_isset(global.close_spread_end)) {
+					int remaining_window = tick_remain(now_ms, global.close_spread_end);
+					if (remaining_window) {
+						/* This should increase the closing rate the
+						 * further along the window we are.
+						 */
+						send_close = (remaining_window <= statistical_prng_range(global.close_spread_time));
+					}
+				}
+				if (send_close)
+					goto release;
+			}
 		}
 	}