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Hi Will,
After our further investigation, we found that the properties which can show in the Word->Info are which can display in the Library Settings->Column list. If you can see the column in Library Settings->Column list, it will show in the Word-> Info.
As there is no official article to explain this situation, based on our test, the reason why “Sensitivity ” and ”Label Setting” column can’t show the value may because there need to cooperate with other Microsoft service like policy, so they have specific function. It should belong to SharePoint property, thus, they won’t be mapped to the document properties. And the column which create by ourselves, like the Version we added value manually can display the value in Word->Info.
In Word application, it can't show the default column of SPO. And we really understand how convenient this feature will be for you, we sincerely suggest you click the Smile icon in Word and give the feedback within application. The related team will notice this requirement directly, it may let them know the requirement of customers so that they can improve our products. However, we still want to thank you for providing this good question for us. We can keep moving on just because the users like you to provide feedback.
We appreciate all your understanding and efforts.
Best Regards,
Ivy