Instructors also check to see if you have completed it.
They allow you access a few weeks prior, but for my last (767) upgrade I was in the middle of trying to sell off 4 cars, rent out my house, close out a tax audit on my business, get a daughter accepted and off to college, etc etc. My personal life was a wreck--nothing BAD, just crazy busy before moving 8000+ miles away. Day 1 of class I had two pro instructors come up and say "hey...are you OKAY? Can you hang with this? Are you READY?" since I had only done about 60% of the LMS.
I assured them that my LMS not being done was a by-product of a very hectic life, and not a sign of indifference, and not to worry about it because by the end of week 1 I'd nail their test and would be just fine. Having done 3 previous programs at FDX I felt pretty confident in how I learn, and I would catch up very quickly with the online training. As expected, the entire course went just fine. My first week of class I probably had to burn time at night than I wanted catching up on the LMS, but got it done and had zero issues in training.
That said, if I HAD struggled on a test, PV, or MV etc I'm pretty sure it would have been pointed out that I showed up for class with those things incomplete. That probably would have painted a picture of me as a student that I probably wouldn't like if I had to do a TRB. As a guy who taught for 10 years in an F-15 schoolhouse I can understand their concern about a guy showing up unprepared, and I wouldn't have liked my approach either had I been the instructor. In this case, I highly recommend doing what I say, not what I did...
What I did do once complete with the program was write up a pretty detailed end of course critique. I suggested LMS be available 60-90 days out, not just (I think) 30 we had a the time. 30 days isn't much for someone moving to an FDA with the long list of things that gotta get done. I have no idea if they made any changes (and I seriously doubt they did) but I tried.
Point is--if at all possible--get it done. They are watching. And its more fun to go out drinking and dining week 1 with your sim partner than doing LMS cleanup.