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?