I have a very mild stutter but it seems to get worse when I'm around a client who stutters severely. Any advice would be greatly appreciated.

I have a very mild stutter but it seems to get worse when I'm around a client who stutters severely. Any advice would be greatly appreciated. I have a mild stutter. It's mild enough that I've never sought any kind of therapy or treatment and don't know proper terminology for my type of stuttering so please forgive my ignorance. It's mostly just getting stuck on sentence starts on occasion. An example would be something like this: "The d...the d...the dog needs to be let outside." It gets worse when I am tired and severe when I am in the prodrome period of a migraine. During a migraine, I get stuck on words throughout the sentence and sort of forget how to say words. The sentence example above becomes "The, um, you know, th...th..th...the thing...it barks, n..n..needs to let, go outside." Anyway, I have this very nice gentlemen who is a client of mine, who stutters. When having a conversation with him, my stuttering increases 10 fold and I have no idea why. It is certainly not intentional. I feel pretty bad about it because I'm not sure if he notices when I talk to other clients that I don't stutter as much and I feel like an asshat. Do any of you know why I do this? Should I explain to him that I too have a mild stutter or should I just ignore it?