I've stored mine for about 4 years (NE Florida, near Jacksonville/Saint Augustine), with very little shade. Before that it was on a dealer's lot down near Lakeland for about a year.
Since I've owned it I've always kept mine closed... well except for a couple of very short periods where I played with leaving it open (like a few days short). I faced the same question you are asking, and finally settled on closed with the idea that it'll reduce the risk of critters getting in... bugs, rodents, etc... I also figured that in really bad wind a bit of water could get blown in, just adding moisture, dust, and dirt to the equation.
I made custom reflectix shades for every window, and I bought those pillow things for the roof vents to keep it shaded inside...and that does help keep the temperatures down a bit.
For reference, I usually go check on it once a week or so...sometimes a couple times a week but often it could be around 2 weeks between checks...when I do check it I go in and have a quick look-see every time. Between uses, it's sits in storage for roughly 1 month to about 3 months, usually.
No so far, I've kept mine uncovered...but after this weekend's hard wash and polish job because it was getting embarrassing to look at (fading and dirt) ... I'm starting to re-think a cover.
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