Location: Dressrosa City Area
Petitioner: Sorrowful Poppy
Reward: Nostalgic Necklace, three Sanji's Cube Fragments, 37,000 Berries
Objective Reward: Perfected Exhaust Ball Plus

To unlock this Side Story, as with all Side Stories in Dressrosa, you first need to defeat Doflamingo and bring peace back to the city. Return to Dressrosa in Chapter 9 and you'll find a bunch of side content to do, including this Side Story. You'll find the petitioner, a green-haired woman, on the south end of the Coliseum, in the far west of the city. She's standing a short walk north of the Square, At Fountain Yoisa Sign.

The petitioner, Poppy, is quite obviously distraught when the Straw Hats walk up. Her boyfriend, Neline, has gone missing. He was a member of the Donquixote Family, and has vanished since Doflamingo was defeated and the marines arrived. She begs the crew to track him down, and Sanji, ever the ladies' man, agrees to track him down.

Poppy's only bit of information is that Neline is on the New Royal Plateau, and that he squeezed through a hole 'the size of a child' to escape capture. Head to New Royal Plateau: 1st Level via Yoisa Travel, then make your way north until you find the first bridge that Franky constructed earlier in the game. Head west from here, as far as you can go, to find a hole for Chopper to enter. It runs beneath a fallen building.

Head east on the other side of the hole. There's a bridge ahead, and a dirt path beside it. Stick to the dirt path and it will bring you to a Donquixote Family soldier named Ekie. Take Ekie and his buddies down and Ekie will point you north, to find another of his friends. You can either proceed through the ruins or use Yoisa Travel to jump to New Royal Plateau: 2nd Level, then backtrack through the Sewer to New Royal Plateau: 1st Level.

The next Donquixote Family goon, Dejide, is in the ruins of a blue building beside the exit to the Sewer in New Royal Plateau: 1st Level. He, too, will attack, though he's easy to defeat. Beat Dejide down and he'll grudgingly point you to Spike Canyon, on New Royal Plateau: 3rd Level.

Yoisa Travel to N.R.P. Sunflower Field and drop your way down the plateau to ground level. At the bottom you'll find Neline, and he'll angrily sic his buddies on you. They're no match for the Straw Hats. Neline will disappear after the battle, and you'll need to share the bad news with Poppy.

(Brook manages to undercut the tension just before you receive your reward, so don't fret the conversation too much.)