During a call with the LH first class booking office in Germany (I'm having difficulties in changing a full first flex fare ticket by one day, as noted in another thread here), they suggested that I call the SWISS first class booking/ticket office in CH. The number they gave me was +41 848 365 365.

I did not call the number, so I can't vouch for it being correct (vs. just another regular booking office), however someone is welcome to try it and report back.