Data point: I mailed my birth certificate (Sex: Nonbinary), driver's license (Sex: X), and deed poll to the Federal Benefits Unit of the US Embassy in London on Thursday, with a request to change my Social Security details (name and sex).
I got my documents back today, no letter or anything.
@ryanc could be a public servant that knew "if i process this, bad things happen for this person" and just sent it back without putting it on the record
@irina My passport lists my sex as "X", updating my social security record isn't going to make anything worse.