I don't think it's as much about hygroscopy as it is about oxidation of materials, and it's not really about the crystalline components of the LD itself, but instead about the facet coatings. The facet coatings may oxidize and make them ineffective or lead to COD, and I suppose they may absorb some water (who knows what facet coatings a company used in making the diodes). But the InGaN, AlGaN, and GaN are all pretty much inert. You'd have to do some major-league reverse engineering to find out what's in the diode, what materials and structure they used in both the crystal growth and processing, and then a ton of major tests on those materials to see what effects exposure to atmosphere will generally have on them, but I'm going to guess that there's not a lot that's going to happen to the actual nitrides in the laser, but that the facet coatings MAY suffer from exposure.
But really, there's almost zero to gain, and there's a lot of risk (mess up the diode when removing the can, exposure to atmosphere, etc.). if the can is fine. If the window is already broken, however, I think you're risking more to leave it on than to try to remove it and save the diode. If the window is broken, it's already exposed to atmosphere, so go ahead and remove the can. Otherwise, there's VERY little extra output power to be gained.
Anecdotally, the window broke out of a PHR diode that I have, so it has been exposed to atmosphere for about 2 months now, and I haven't noticed any degradation in performance.
ETA: Also, I doubt any companies so far have taken the time/energy/money to investigate if exposure to atmosphere will have long-term deleterious effects on the lasers, so I'd be willing to wager that them putting a pan on the diode instead of the open-can approach is from them taking the same "better safe than sorry" approach to the problem that I take, to just leave it unless it's broken anyway.