I'm hiking 65 kms in 6 days in the mountains of Peru. Boots are highly recommended.
I'd definitely agree with BHZ. Trail runners will immediately solve your problem. If you don't have ankle stability issues that is. I did 2 treks in Peru this May, the 5 day Salkantay(70km up to 15,500') and 6 day Ausangate(92km up to 17,000') Trek in my trail runners. Boots are not necessary unless you're going when it's snowing or mountaineering. We actually had a blizzard day 5 of the Ausangate trek and I was fine in them. Wet feet that day but they were dry by the morning and surprisingly stayed dry in our snowy trek from camp to rainbow mountain.
The Peruvians made for a very humbling experience haha. Hiking up crazy steep terrain twice as fast as us but in slip on sandals.