Neighbour stutters which causes me to stutter.

Neighbour stutters which causes me to stutter. I have tried looking this up but cannot for the life of me find anything. My new next door neighbour has a very bad stutter and they're nice to talk to but I caught myself on more than one occassion stuttering when speaking to them for the first time. I wasn't doing it on purpose but it just kept happening. I don't have a stutter and as far as I'm aware I never have. The more he spoke when I needed to answer I'd stutter back to them which I noticed increased their stutter which made me stutter back to them more. I had to really focus on not stuttering to stop it. It was a really embarrasing situation and I worry they think I'm doing it on purpose. I didn't bring it up because it was awkward. Why did I start stuttering?