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Nope - you do not need to republish the label policy if you only change the Description or remove the Content marking Header from an existing sensitivity label. The changes to the existing label propagate automatically - just allow up to 24 hours for replication and for Office clients/services to reflect the changes.
To make the change, go to Microsoft Purview > Information Protection > Sensitivity labels, select the existing label, choose Edit label, and update the Description fields as required. Proceed to the Content marking section and remove/disable the Header configuration. Leave the existing Scope and Access Controls unchanged, review the configuration, and select Save label.
The main consideration is existing content. Removing the Header from the label definition does not automatically remove headers that were already applied to existing files or emails. It prevents the header from being applied when the label is subsequently applied or reapplied. Therefore, test both newly labeled content and existing labeled content.
Also allow for propagation and client caching. Changes may appear within minutes but can take up to 24 hours. If a client does not reflect the change, restarting Office applications or signing out and back in can help. There is no need to create a new label or republish the label policy for these changes.
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hth
Marcin