Microsoft Authenticator on iOS: iCloud backup does not restore accounts on new iPhone

I need to migrate my accounts from Microsoft Authenticator on an old iPhone to a new iPhone, but the restore process is not working.

Scenario

Microsoft Authenticator is still working correctly on my old iPhone, and I can access my accounts from that device.

I installed Microsoft Authenticator on my new iPhone. iCloud is configured on both devices, and Authenticator indicates that a backup is available/enabled.

However, after installing and setting up Authenticator on the new iPhone, the accounts that are available on the old device were not restored to the new one.

This is particularly important because I currently depend on Authenticator on the old phone to access some of these accounts, and I do not want to delete, reinstall, or modify the app on the old device in a way that could cause me to lose access.

Environment

• Microsoft Authenticator
• iOS / iPhone
• Old iPhone: Authenticator is still working and contains the accounts.
• New iPhone: Authenticator is installed, but the accounts do not appear.
• iCloud and Authenticator backup are enabled.

Troubleshooting already performed

I have already followed Microsoft's official instructions for backing up and restoring Microsoft Authenticator credentials, including the documentation for:

• backing up accounts in Microsoft Authenticator;
• restoring account credentials from Microsoft Authenticator.

I followed all the steps in the documentation, but the accounts still do not appear on the new iPhone.

I also contacted Microsoft Support about this issue. After reviewing the situation, the support representative advised me to post the issue here so that Microsoft Authenticator specialists or moderators could provide further assistance.

Question

What is the correct and safest procedure to recover or transfer my Microsoft Authenticator accounts from the old iPhone to the new iPhone when iCloud backup is enabled but the restore process does not recover the accounts?

I specifically want to keep Microsoft Authenticator on the old phone completely intact until I have confirmed that all accounts are working correctly on the new device, because for some of these accounts I currently do not have another way to authenticate.

Thank you.