Maybe, while the car is running, the PV panels could be used to drive a high-efficiency fan, aimed backwards, externally, to boost speed and improve mileage. It might produce a 1-2 MPG improvement at highway speeds over the lifetime of ownership. Big oil is probably preventing this simple improvement.
The panel is reportedly 50 Wp. In optimal orientation towards the sun at my location that would generate about 50 kWh a year. Solar panels flat (like the roof of a car) supposedly do 87% of that, so 43.5 kWh. You can do the math what that costs in your location, for me it's about € 8. That is what you could save in an optimal situation. You'll have to park your car in the sun whenever it shines and obviously you'll get more (I think even double) when you're living more towards the equator and less when you're living more towards the poles. So economically I don't think the solar panel is a good option.
Just to help keep various threads on this topic together, here's a link to Michael Cline's solution from June of 2013, which was pretty slick. and vallesj's very nicely documented setup from a year later. It's interesting that this thread was started about a year before Michael Cline implemented the idea, and somehow the connection between the two threads never got made. But the two implementations linked above did demonstrate that it is practical to use the solar roof to maintain the aux battery (this interests me right now, as with COVID the car is being used less), and even without completely losing the vent function).