Shoes for hiking

I’ll be in Japan for three weeks in September/October, and will be doing a couple day hikes on the Nakasendo, plus an overnight hike on the Nakahechi section of the Kumano Kodo. I want to travel light, and not bring more shoes than I need. Are light sneakers adequate, or would it be much better to have hiking shoes, even if they’re heavier than I’d want for the rest of the trip? Are the trails very rough, or very slippery in the rain?