I've never used any treatments on my strops. Some are vintage, and some were bought in the last ten years. I would think any type of leather conditioner, oil, even soap would change the look and feel of the leather. I've restored vintage strops, once clean, I treated them with a very, very small amount of neetsfoot oil. The leather darkened, and the draw increased.

Now, I only rub them down with my hand, and it works great...! The natural oils in your hand should be enough, At least in my experience it has.....