Today's daily task for the Heavy Metal event requires that you complete 3 stages of a Public Quest.
It does not specify that it has to all be the same one or that they have to be of your appropriate tier. You can solo an easy phase 1 3 times and it will count. There is a chapter 13 PQ in the Saphery zone that requires you to click on 8 stones to complete. I had a guildmate solo a chapter 1 PQ for his daily task even though he was level 31 and it counted.