Hello,
What version of Adaxes are you using and how do you create remote mailboxes? Starting from Adaxes 2016, creation of remote mailboxes is supported out of the box. If the remote mailboxes are created by Adaxes, there should be no replication issues as Adaxes passes the DC it uses over to the Exchange Server. The Exchange Server uses the DC passed by Adaxes for creation of a remote mailbox. Thus, if the Scheduled Task is triggered on a certain user, this means that it has already been replicated to the DC used by Adaxes, and also means that the user account should be available to the Exchange Server, since it uses the same DC.
If you are creating remote mailboxes with the help of the Enable-RemoteMailbox cmdlet, you can use its parameters to pass the DC used by Adaxes to that Exchange Server. To get the DC used by Adaxes, you can use the following code in your script:
$domainName = $Context.GetObjectDomain("%distinguishedName%")
$rootDSE = $Context.BindToObject("Adaxes://$domainName/rootDSE")
$domainControllerFQDN = $rootDSE.Get("dnsHostName")
Then, you can use the DomainController parameter of the cmdlet to pass the DC over to Exchange:
Enable-RemoteMailbox "%distinguishedName%" -RemoteRoutingAddress "%samaccountname%@someone.onmicrosoft.com" `
-DomainController $domainControllerFQDN