Has anyone proven if jumping is faster, slower, or the same speed

It seems to me based on top runs I would say most likely not.

So the same speed

jumping up slopes is faster afaik

up slopes, yes it's faster, but idk about just a flat surface.

Ok

your speed wouldn't change if you jump on flat land

So does that mean on flat surfaces if normal speed and in slopes faster?

Yes, and down slopes i'm not sure. However, when you jump while on down slope you will have more horizontal speed then if you were to just run down it. (i think)

that is true i use this in my runs all the time