# homotopy theory – Does lifting correspondence hold for principal bundles too?

Let $$P$$ be a (nontrivial) principal bundle over the base space $$mathbb{R}^4$$ and fibers diffeomorphic to $$SU(3)$$.

Also assume that $$P$$ is equipped with an Ehresmann connection.

Then, for for any two given points $$x,y in mathbb{R}^4$$, all paths that connect them are path-homotopic.

Does this imply that the horizontal lifts of all these paths onto $$P$$ that start at the same point are also path-homotopic? Or at least can I assert that all such horizontal lifts end at the same point?