On complementation and $L$-embedding of Lipschitz-free spaces in their biduals

A closed subspace $E$ of a Banach space $F$ is \emph{$L$-embedded in $F$} if there is a projection $P\colon F\to E$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in F$. We show that if $X$ is a proper, geodesic metric space with locally finite $\H^1$-measure, then its Lipschitz-free space, denoted by