The issue I caution about using synthetic oil, is that it will clean and loosen all the dirt and grime inside a diff. Especially on one that has never been changed. So what ends up happening since there is no filter to filter out this dirt and grime, it then gets recirculated into vital wearing surfaces.
Hopefully no damage is done, but I would drain and refill the diff with conventional 80w90 hypoid gear oil from a MB dealership.

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