This character story becomes available in Chapter 2, after completing a total of 10 requests (regardless of rank, normal, or repeatable).
[-] Consult sympathetic members
First, talk to Levy to obtain the New Picture of Wendy.
[-] Find Wendy!
Next, find Wendy on the bridge to the western side of the town. Talk to her to hand over the new picture and complete the story. Her max character rank will increase from 3 to 6 as well.
This character story becomes available in Chapter 4. This triggered for me right after completing Juvia’s second character story “Isosceles or Love Triangle?”. Levy will approach Wendy and after that cutscene, talk to Wendy to start the quest. Wendy is the required character but you’ll be able to select 3 more allies to join her party.
[-] Find the medicinal Ingredients
First, Fast-Travel to Akane Beach and head to the marked gathering point to the west. After picking up the Suspicious Herb, travel to the Boundary Forest next.
[-] Go give the Ingredients to Porlyusica
Head to the cottage in the northwest section of the forest and approach it to trigger a conversation with the old healer. After the conversation, the quest will be complete and Wendy’s max character rank will increase from 6 to 9 as well.
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