Is it a vinyl top or a fabric top?
My Boxster is fabric...so I do have experience with fabric tops.
It is a fabric top.
Alright.
The very first thing I'd do is properly wash and then protect/water proof the top so you're at a known, good starting point. I used 303 products to do so and the results are pretty dramatic. Not only does the water bead, but it sheets right off!
303 (30520) Convertible Fabric Top Cleaning and Care Kit
https://a.co/d/2Qin0mjI used the method found here:
https://youtu.be/e-efGzzefREAs far as touchless washes go, they generally use harsher chemicals (because touchless! ) and I'm not convinced the rinse is good enough to get all the harsh soap that may have seeped in to the top.
I would suggest a spray wash of some type and then applying a VERY good hydrophobic product to the car so it stays clean longer.
The product I use is quite spectacular in this regard as I spend $$$$ every year trying the latest and greatest products hoping one of them knocks my "Top Go To" products out of my list. The car looks darn near washed after a rain shower. I've sent a bottle to a forum member to try so perhaps he can comment to.
My current greatest, glossiest (instrument measured) topper/wax/sealant/quick detailer is Shinee Wax.
https://carzilla.ca/products/tacsystem-shinee-wax-spray-detailer-1000mlAnd yes, it will be fine to be left outdoors AFTER you treat the roof.