Heres my theory....

To keep a strop in good order one must occasionally treat it with some form of conditioner. to keep it spft and supple.

Leather is just cow skin,

mitchels wool fast soap is a great for keeping the skin soft.
so would occasionally (once or twice a year as you would if you were using normal strop paste) dabbing a bit of lather from MWF soap help to keep the leather soft,
Or does something in the tanning process change the make up of the leather meaning that treating a strop with MWF would be pointles/harmfull?

Just an idea......