I had never heard this version of the myth before. The "myth" I have heard, is that if you do a search on say Travelocity for example and then go and search Expedia or another site, you will see lower prices. This myth, to me at least, is a little more plausible, although in my own comparison I have never seen a major difference in prices to get me to convert from one site to the other. Obviously each of these sites competes against one another for bookings and traffic, so if they know you are "shopping around", by checking if a cookie exists from another site, it is in their best interest to offer lower prices.
Summary: No. "Flight prices are most likely impacted by three factors: complexity of airfare pricing, high demand for the same flight and cached data."