I recently purchased my first razor and strop. My concern is that the leather on my strop is too stiff. I have read on here about the need to dress a strop, etc. and about how usually there is little that needs to be done.

My strop is an Illinois 127. It feels stiff like a new baseball glove. Should it be more supple? I have some good leather conditioner that I have used on leather. Would this be appropriate to use?

Tandy Leather Factory - Dr. Jackson's Hide Rejuvenator

Also, how supple should the leather feel? Like a well broken in glove? I haven't seen a good example of a strop, so I don't really have a good frame of reference.

Any help for a newbie would be appreciated.