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Your Customer can see generic Customer Prices from their SuperSite 2 even though you have set Specific Pricing for them

If you have set specific pricing1 for a particular Product/Service for a Customer of yours, then the same would not be immediately visible to him/her when he/she logs into their SuperSite 2, while buying an Order of this discounted Product/Service.

However, at the time of Payment, your Customer will be presented with the Specific Pricing you have set for him/her.