I don't have any documentation to fall back on, so this is just my thoughts.
I would leave it up fully. I do know some that lower theirs back down onto the bars. Either way, I don't think it should hurt anything. Personal preference of mine would be to have it all of the way up and have less possible wrinkles in the tenting that might lead to unintentional leaks. Highly unlikely, but taut is better than loose in this case. If it is resting on the door, that might cause it to warp some over time. And one are we don't need help with is warped doors and frames.