There may be some data behind that 4-to-6-months-before-travel timing, but it's too broad an average to be useful. You need separate statistics for your specific origin/destination, the month of departure and ticket class, at the very least. Even then, I wouldn't bank on it. Whether your home airport is a relatively competitive air-travel market is a major factor. If you are wedded to British Airways by preference or due to limited options, that reduces your chances of finding a good deal. These days it seems everyone is looking for something better than a standard coach-class ticket, so there are a lot of people chasing premium tickets.
I started traveling to Europe 55 years ago. There are always airfare sales being advertised, though in the old days you had to watch for newspaper ads. I have never, not once, snagged a good fare after hearing about a sale. The sales always had restrictions (most often date restrictions) that didn't match my travel plans. What has worked for me is just what you are doing--tracking fares for a decent period so I know more or less what is normal. Doing that doesn't guarantee a bargain; there may not be any real bargains (especially in June). It does reasonably assure that you'll spot a price dip if one occurs and be aware when the current price is significantly higher than it has been.
Rather than relying solely on website alerts, I'd also check manually at least once a day. Checking more frequently might pay off; you just don't know when an airline may adjust its prices, and there may not be many tickets available at a newly announced price. A few years ago I watched a coach ticket into Munich and back from London bounce repeatedly between $800+ and $1300+. It was changing at least every few days approximately 3 months before I planned to depart. My home airport is a relatively competitive one, with lots of airlines offering transatlantic flights, which may lead to fare volatility.
I assume you're looking at a multicity ticket into Rome and back from Paris, correct? What is your home airport?