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This is the SOP at SFO. You're in the BA lounge if you are flying BA or LAN, the CX lounge if flying CX, and I would imagine if one were to push them, they would indeed play the capacity card. The CX lounge is quite large, so it isn't a plausible gambit. I don't know for sure about the JAL lounge close to the CX lounge, but there again I doubt anyone would willingly want to go there anyway.