FFXIV's entire 6.X storyline will center around the thirteenth shard, as confirmed by Naoki Yoshida. Players have guessed as much since 6.1, Newfound Adventure, teased the appearance of a mysterious voidsent, but Yoshida's interview confirms that the 6.X arc will focus on what happened to the thirteenth shard, as well as the memories of feelings of those who were once human, but are now monsters.
Yoshi-P says in a new Famitsu interview that FFXIV Patch 6.2 and the rest of the 6.X patches will revolve around the Thirteenth Shard in its story. He also says that Patch 6.5 is where you'll see a glimmer of what's to come in 7.0!
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