Let $P \xrightarrow{\pi} X$ be a smooth $G$-principal bundle with a connection $A$. It is known that for two smooth curves $\gamma, \delta$, there are respective horizontal lifts $\tilde\gamma, \tilde\delta$. A result from the Lee states that any continuous homotopy between $\gamma$ and $\delta$ gives a smooth one, if $X$ does not have a boundary. Further, there is the continuous homotopy lifting property.
In this setting, is there also a smooth homotopy lifting property? This answer does not apply here, as local triviality of the bundle rules out the counterexample given there.
It seems to me, that the construction for the horizontal lifts should also work over the smooth homotopy, simply going from 1 to 2 parameters. However, I cannot find this result anywhere. Am I missing something?