Weird, given the number of people who said this bugged out for them, I figured it was a fool's errand to try do this since I wouldn't be able to turn it in. But I did all the tasks, went back to the guy on the nexus, and it marked completed for me.
Maybe the order you do the tasks matters? I fought the fiends first, then did the kett base, and did the remnant last. I also ran around not killing the destroyer until all the remnant had spawned. Drack and Jaal were still fighting it though, so I guess I was lucky they didn't accidentally kill it before the last of the remnant spawned. But, it was still fast enough to beat Andraknor's time.
I didn't go back to him immediately though, after doing the three tasks, I was 82 points away from next level, so I just drove to the nearest group of Kett and killed them to get the next level. But after I did, I did immediately go to the Nexus and turn it in. So not sure what I did exactly for it to work for me, but that's what I did.