I completed both levels “Cortana” and “Halo” on Legendary today in one session, and while the mission select screen now shows that I’ve beaten “Halo” on Legendary, it does not show the same for “Cortana.” This also means that the achievement for completing the campaign on Legendary did not unlock.
My friend and I just completed halo 3 on legendary and ran “Cortana” twice and it still isn’t showing as completed either. I hope they fix it soon 
Hey 343, are you ever going to fix this problem? I’ve run into it at least 5 times!
I got it to work when I ran it on normal, killing most of the flood on the way, and then clearing the way for the Arbiter to join me in the landing area next to the end.