Hi all, Public Folders and Hybrid mode. Not really a hybrid mode cause all mailboxes reside in the Cloud, also PF Mailboxes. First set Default PF for all mailboxes. 
Get-Mailbox -ResultSize Unlimited -RecipientTypeDetails usermailbox | set-mailbox –defaultpublicfoldermailbox o365PFmailbox
When sending email from outside organization sender will get. 550 5.4.1 Recipient address rejected: Access denied. or 550 5.7.13 Anonymous users are not permitted to add contents to mail enabled public folder, authenticated required. And fixes are.
  1. Modify Allowed domain to Internal Relay, instead of Authoritative. This is cause Directory Based Edge Blocking (DBEB) block external senders cause the address isnt't inside EOP users.
https://docs.microsoft.com/en-us/exchange/mail-flow/accepted-domains/accepted-domain-procedures?view=exchserver-2019#CreateAcceptedDomain With Exchange Online powershell. Add Anonymous access to CreateItems: 
 Get-PublicFolder -Recurse | Add-PublicFolderClientPermission -User Anonymous -AccessRights CreateItems
And as an advice, raise PF limits to prevent errors when migrating. 
Set-OrganizationConfig -DefaultPublicFolderProhibitPostQuota 10737418240 Set-OrganizationConfig -DefaultPublicFolderIssueWarningQuota 9663676416

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