I did this on a bard before they added these to get rid of some scales. I was surprised to see the update on that character early this morning for the achievement. Some recent stuff apparently will show a task update if it was done before these achievements were added.
[Thu Jan 02 22:16:53 2020] Exterminator Sahizz says, 'Nice work. Here iss your money. Now leave me. I have work to do.'
[Thu Jan 02 22:16:53 2020] Your faction standing with Legion of Cabilis could not possibly get any better.
[Thu Jan 02 22:16:53 2020] Your faction standing with Cabilis Residents has been adjusted by 1.
[Thu Jan 02 22:16:53 2020] Your faction standing with Scaled Mystics has been adjusted by 1.
[Thu Jan 02 22:16:53 2020] Your faction standing with Crusaders of Greenmist has been adjusted by 1.
[Thu Jan 02 22:16:53 2020] Your faction standing with Swift Tails has been adjusted by 1.
Now if it only went back far enough for my dark elf rogue and the Lyda Nasin achievement.