I managed to book the following for two persons traveling in April this year for a little less than 3500EUR per person:
MXP-ZRH-SIN on LX
SIN-AKL on SQ
AKL-JFK on NZ 2
JFK-FRA-MXP on LH

A few notes for people who are looking to do this. I started based on information here in travel-dealz eu /deal/business-class-around-the-world-italy/
expterflye trial was really useful to see availability on specific classes on specific flights. Without it I would have gone mad as google flights would show you the price until the very last step and you don't have visibility into which leg(s) mess up based on seat availability.

In general I only managed to get reasonable prices for trips starting with LH group flights. LH and LX were fine but SQ and EVA Air direct flights from MXP that had availability in the cheapest Business class fares never priced correctly. I only looked for eastbound flights so westbound might be different.

Air NZ look for booking class Z availability, there is one that is cheaper I think but there was never any availability. AKL-JFK nonstop availability started end of april, May was much more relaxed.

On LH you are looking for P fares mostly. Ticketed classes were different though: LH group: Z, SQ: U and NZ: J.

On SQ look for availability in D.

Then when google flights priced it, it also told where to buy it for which price. Not all of them priced it the same afterwards.