Does lifting correspondence hold for principal bundles too?

Question

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?

Answer 1

Let P be a (nontrivial) principal bundle over the base space R^4

All principal bundles over R^4 are trivial because R^4 is contractible.

Or at least can I assert that all such horizontal lifts end at the same point?

No, because the curvature of the connection on P can be nonvanishing, in which case you can find two homotopic paths whose horizontal lifts end at different points.