TL;DR:
Yes, for at least 20 minutes after the reply arrives.
You shouldn't move any replies from your prospects to other folders or delete them from your inbox for at least 20 minutes.
Woodpecker uses those 20 minutes (that's the refresh interval) to check and analyze the new message in order to assign it to a specific prospect and campaign. The reply will then be shown in your Inbox in Woodpecker.
Later on, you are free to move it to another folder, archive it or delete it, at your leisure.
Why is it important?
The sequence of follow-ups will be stopped once the RESPONDED status is correctly assigned to your prospect. If you move or delete the reply before your prospect's status gets updated, next emails in the queue will be sent - and this is what you want to avoid.
Please make sure no forwarding or labeling has been set up on your mailbox. Besides, check if IMAP has been enabled. Otherwise, Woodpecker won't be able to fetch your responses.
Have a look at our article Deleting emails from Woodpecker Inbox if you want to learn how to remove emails from your account.
Don't worry, they won't be deleted from your mailbox server unless you choose this option!