After a bit of thought I have a sketch of a very simple case. Suppose that $B$ is a 3-polytope; let $B^*$ denote its dual and consider the graph $G$ associated to the 1-skeleton $B^*_1$ of $B^*$. Now vertices of $G$ correspond to faces of $B$, and edges of $G$ correspond to adjacent faces of $B$. So if $G$ admits a Hamiltonian path then we can use it to get a (nearly?) optimal peeling for $L$ appropriate.