If I'm not mistaken, CD timers on missions that you can pick up start from when you pick them up, not when you drop or complete it. Kind of like the dailies
I have noticed on some daily missions (not his one, not tried this one yet) that sometimes when I drop a daily it will start the 20 hour counter anyway. Last time this happened was a few patches ago, with the officer report daily. At the time, I assumed it was working as intended and I resolved to never drop a dailly if I could help it.