A Lipschitz refinement of the Bebutov-Kakutani dynamical embedding theorem for connected Lie group actions

Let $G$ be a connected Lie group equipped with a compatible right-invariant metric and let $G$ act continuously on a compact metrizable space $X$. We prove that $X$ admits a $G$-equivariant embedding into the right argument shift action on the space of one-Lipschitz maps from $G$ to $[0,1]$ if and o