They generally can only rise in that short timeframe.

As the day of departure draws nearer, fares that have minimum advance purchase requirements sequentially get closed off; this alone would generally always cause the cheapest-available fare to rise, stepwise, as time marches on.

Even if some people cancel their tickets, or change to a different flight, in the 7 days before departure, this is highly unlikely to result in the revenue management department opening up new seats at a cheaper rate. Load factors are relatively high at the moment, so it's unlikely that a flight would be so empty at T-7 that they decide they need to drop the price in an attempt to get more bums in seats.

I don't typically monitor prices in the last 7 days before departure, but I would be surprised if the price dips on a particular flight in more than a handful of cases.