Is warm clothing required in Zermatt during June?

Can any expert advise whether we need warm clothings in Zermatt or Klein Matterhorn during the month of June?

Not down in the village but up in the mountains for sure. Not a bulky downjacket but good wind and waterproof protection

In the mountains, you need to carry rain gear and a warm sweater any time of the year.

Early morning temperatures at Kleinmatterhorn may be around zero degrees centigrade.

Thanks!