(It is a .docx document, not a template.)
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I am seeing the same thing from that document.
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This is in 365, Version 2309 updated a couple of days ago. (Windows 11)
This is on all odd-numbered pages, not just the first one. The ruler stays the same on the even- and odd-numbered pages.
(The paragraph formatting noted by the black squares is keep lines together, which should have nothing to do with this. The document is not in compatibility mode.)
Yes, you are correct it should not do this.
This forum is a user-to-user support forum and the people who can fix these do not see posts here.
Use the feedback mechanisms to report this bug to Microsoft Developers.
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Note that many of the "updates" or "repairs" to the desktop application Word are shared in all versions of Word from 2016 on. They share much of the same code.