Given the way fares and bucket availability are distributed through GDS channels, such per user fare manipulation would be trivial to detect. I can't imagine an airline would risk potentially millions in class action lawsuits to play such games. Check third party sites like ITA matrix if you really believe there is an issue.

Note that if you are searching for 3 tickets, you need to be sure you always search for that number as there may be fewer than that number of seats available in a fare bucket. Searching for a single ticket may give a different price than 3 tickets if there are only 1 or 2 seats left in the bucket as it will push the fares into a higher bucket. Looking at the number of seats open on a seat-map is completely meaningless. You are searching during a very high demand time for flights and airlines are hesitant to sell too many seats at too low of a price as they know those seats will command higher fares based on past sales history.