There are plenty of reasons to book early, but getting a low fare is not one of them.
Nowadays only a few airlines will price match their latest fare when you proactively call and ask, and in all cases it is in the form of a voucher (gift certificate) rather than money.
Long long ago, airlines wanted more cash under their belts (in their coffers) sooner so they offered to price match when earlier buyers did the research and found a lower fare for the "exact" same flights. Nowadays most airlines found out that they ended up with more cash altogether when they did not go for more cash sooner, hence the end to the price matching.
As far as a "six week rule" goes, it's a question of "who will blink first". The passenger for fear there won't be seats left, or the airline for fear seats will go empty.