My advice is to set up fare alerts for several acceptable itineraries on your preferred airfare website. I use Google Flights, but there are several others. Being a non-trusting sort, I also try to do a manual check every day or so while I'm flight shopping. It's not a human being tweaking the fares, it's a computer program, and sometimes it produces very odd short-term results, at least for some routings. I've seen $500 upward spikes followed 24 or 48 hours later by $500 declines. A few days ago, a flight I was watching earlier in the year went from under $900 to over $5000. Wacko. A brief fare drop could happen at any time.
It would be a good idea to expand your search to include one-stop options, just in case. Non-stops are popular, so if people keep buying tickets on the non-stops to Rome, it's possible there will not be much of a drop, at least not early enough to help you. I assume you do not want to still be flight-shopping in late August, right?
As already noted, a bit of date flexibility can also be helpful. I've seen price differences of $250 or more (round-trip) as a result of shifting one of the flight dates by just one day.