Let $\pi:E \rightarrow M$ be a principal $G$-bundle for a Lie group $G$. Let $\omega$ be a connection on the principal bundle. It is a well known fact, that if a path $\gamma:[0,1] \rightarrow M$ is path homotopic to $\gamma': [0,1] \rightarrow M$, then parallel transports induced by $\omega$ along $\gamma$ and $\gamma'$ coincide.
For the definition of path homotopy, please check (https://ncatlab.org/nlab/show/path+groupoid#definiti)
My question is the following:
Let $\gamma$ be path homotopic to $\gamma'$. Let $p \in \pi^{-1}(\gamma(0))= \pi^{-1}(\gamma'(0))$ and let $\tilde{\gamma}_{p}:[0,1] \rightarrow E$ and $\tilde{\gamma'}_{p}:[0,1] \rightarrow E$ be the unique horizontal lifts of $\gamma$ and $\gamma'$ respectively through p, induced by the connection $\omega$. Is $\tilde{\gamma}_{p}$ always path homotopic to $\tilde{\gamma'}_{p}$?