I apply really tiny amount of neatsfoot oil in October and April - basically only at winter time (winter is 8 month here ) on all my strops, humidity in summer time in Moscow relatively low, and at winter Russians heating their homes like there is no tomorrow. At winter you can have 7-15%!!! humidity in your house, which is definitely bad for strops. That’s it. I don’t bother even with palm rubbing.

In regards to shell cordovan, I have the same strop like yours, I found that no any oil required.