A lot of stuff JP plays relies on open strings and specific layouts on the fretboard. I'd be surprised if he went through the pain of adjusting any of that.
But, the guitar is always the instrument that is affected the most by downtuning. The reason for that is, guitar strings are specifically designed to be under the tension that achieves the intended pitch. So, tuning a low E string to Eb isn't the same sound as a string designed to be tuned to Eb (is there was such a thing).
So, I certainly don't have perfect pitch, but I can always tell a downtuned guitar because the timbre of the strings sounds off.