According to the Item Recovery article from SE Support Center, opening a ticket through them is a valid path towards item recovery. They go so far as to describe the steps to follow in order to facilitate the recovery:
You can place a GM call from in-game using the /helpdesk chat command, or contact a SQUARE ENIX Support Center at the bottom of this article. Simply select "Additional Assistance," then select "Next," then select "Next," then choose a contact method.

I opted for opening a ticket, as opposed to calling a GM in game because --as far as I know-- the only true choice when calling a GM in game is by selecting an option saying that someone is violating the Code of Conduct or something like that. This is 100% not that. This is a missing item (that I probably dropped without noticing.)

Four days later (out of the 7 days recoverable window), they respond with this:
We apologize for any inconvenience you have experienced in-game. Unfortunately, the Support Center is unable to assist with any in-game issues. The Support Center can only assist with account and technical issues, such as an inability to log into a Square Enix account or trouble updating the game client. For assistance with an in-game issue, please contact a Game Master. To contact a Game Master please type “/supportdesk”, which will bring up the menu to contact a Game Master for further assistance.
So my only option was to open a false GM ticket, claiming that someone was cheating or harassing in game, to speak to someone about the lost item. The GM was amazing, friendly and helpful (even if in the end the item is gone, this is true: Zeigunav felt like he was hearing me and going to see what he could do to help.)

My steps have been as follows:
  • Website says to open ticket, I open ticket.
  • 4 days later, response to ticket says I shouldn't have opened ticket.
  • Call GM by falsely claiming TOS violation (only option).
  • Begin item recovery process.

Is this how the system is supposed to work? None of those steps feel right except the last one. Can someone look into fixing this for others going forward?