Hello,
We apologize for the issue you are facing with updating Rank Math PRO. It seems like there might be a problem with the current installation or the update process.
To manually update Rank Math PRO, please follow these steps:
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Download Rank Math PRO: You can download the latest version of Rank Math PRO from your Rank Math account area. If you are unable to find it, please ensure you are logged in with the correct account that has the PRO subscription.

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Upload the Plugin to WordPress: After downloading the ZIP file, go to your WordPress dashboard and navigate to Plugins → Add New. Click on Upload Plugin.

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Install the Plugin: Choose the downloaded ZIP file and click on Install Now. If prompted, select Replace current with uploaded.

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Activate the Plugin: Once installed, make sure to activate the Rank Math SEO PRO plugin from your Installed Plugins list.
If you continue to experience issues, please reply to this ticket with your site URL, and a human agent from our support team will assist you further. Thank you for raising this concern!
Sources:
How to Update Rank Math Manually » Rank Math
How to Upgrade to Rank Math PRO » Rank Math
This is an automated reply from Rank Math Support Bot. If you need further assistance, please reply to this topic and a human support agent will reply to you shortly.