Hey @user2074​! Daniel from Customer Support Engineering here 🔧 

You can achieve this by enabling the option "Prevent visitors replying to closed conversations" under Messenger > Setup > Messages > "Control inbound conversation volume" 👍 Hope it helps!

Hello, @Daniel M15 I have a follow-up question. I see this is a setting in messenger, but will this apply to both chat and email conversations? Or just chat? 

Thank you!