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The original look of the dash is quite dull and this is done purposely to reduce glare. Perhaps the shinier parts are due to some dressing applied by the previous owner. You might want to try cleaning it with a soft nylon brush and Meguiar's Quik Interior Cleaner and see if things even out. Meg's QID adds no gloss at all and no odors either. The dash also will get shiny if you rub it too hard with a microfiber so be gentle. This may also be the cause of what you are seeing now.
I took the liberty of looking at a number of RX350 dash pads before coming to my conclusions about the appearance of mine. I believe that although the dealer didn't offer suggestions (other than dying the dash pad) I think it's possible that their own detail shop may have contributed to the issue with the products they use. I have Meguiar's products in my garage and if the 303doesn't maintain over time I may take a step back and clean as you suggest. Right now it looks very good, with the overall sheen matching the plastic on the door panels next to the glass. I appreciate the warning about the microfiber. Never really gave that much thought but certainly dont' want to create any problems.