donkey i think you need to find a new dealer.
if one RTM it says on 8-19 and 20:
https://www.mymazda.com/MusaWeb/pdf/manuals/2011_CX9_OM.pdf
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Mazda recommends Mazda Genuine Parts
and Castrol (U.S.A and Mexico). For
optimal engine performance, there are
certain types of engine oils and filters
suitable for your vehicle. Please consult
an Authorized Mazda Dealer.
Recommended Oil
Use SAE 5W-20 engine oil.
Oil container labels provide important
information.
A chief contribution this type of oil makes
to fuel economy is reducing the amount of
fuel necessary to overcome engine
friction.
Only use oils “Certified For Gasoline
Engines” by the American Petroleum
Institute (API). An oil with this trademark
symbol conforms to the current engine
and emission system protection standards
and fuel economy requirements of the
International Lubricant Standardization
and Approval Committee (ILSAC),
comprised of U.S. and Japanese
automobile manufacturers.
The quality designation SM, or ILSAC
must be on the label.
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so i actually misspoke earlier. mazda does not specifically recommend conventional. it only recommends 5W-20 with the proper quality and performance specifications listed above.
what i suspect comes in the engine, from the factory, considering it is a ford engine, is a semi-synthetic. ford loves its semi-synthetic and i suspect that was what it was developed to use.
My dealer tells me to use synthetic and warns they can void the warranty if I don't (yes i have all my services done at the dealer while the car is under warranty to make my life easier). I would find a new dealer.