I tried to search but could not find answer. Please direct me to the link if this has been discussed before.

Flying in Coach on a ticket bought from United:
SFO-TLV with United
TLV-ZRH with Swiss (stay in ZRH for a few days)
ZRH-SFO with United
I have Star Alliance Gold and United Gold
Ticket is paid for (not miles) and booking class for Swiss (according to ua.com is L)

Question:
How much luggage I am entitled check on Swiss?
Swiss website says none and I need to pay for a bag 23 or 32 lbs (different prices)
UA website says I get one 50 lb free bag
Called Swiss and was told (upon giving my booking number) that I am entitled to one 50 lb and one 70 lb bag. I asked her to send me a link or to send me a confirmation of this and she says she cannot and that this is what it shows.

Needless to say, not sure who to trust
Any personal experience with a middle segment in Swiss bought with United reservation?