I also have a 4003 that I bought back in 2011, and I've left the PU cover on that one as well. I bought it from our local music store. The owner talked about how the cover was important to the sound and a little tricky to remove/replace. I figured if that's how it's designed... then who am I to mess with it?I’m leaving the cover on as well. I’m not used to playing with one, but it’s a weird kind of novelty playing around it. I do find it to be a useful position reference when I’m playing between the pickups.
Plus I'm great at taking things apart...and not so great about putting things back together.
And I'm pretty lazy.