No. You only have to have it in your inventory. That is the ONLY requirement aside from hard mode. I've tested quite a few of these "theories." First, I do normal every week before ANY other activity for it. The time I got the crux? I didn't have any crota weapons, armors, emblem(I deleted it because I had no room for other shit), and only the glowhoo in my inventory(didn't wear it as my Arctic proxy shader is MUCH better looking to me. Ice on my armor, baby). Again, The only requirements (that I could test) is that you do it on hard mode(more chances if you do it on hard only. double drops because you're not getting normal rewards, you're getting hard mode rewards "twice" except for deathsinger...) and to have glowhoo in your inventory at least. I've got mine with those measurements. Don't worry too much about requirements. It's not that important if you're trying for it. Just have glowhoo with you. Wear it if it makes you feel better