  Grigni and Hung~\cite{GH12} conjectured that H-minor-free graphs have
$(1+\epsilon)$-spanners that are light, that is, of weight $g(|H|,\epsilon)$
times the weight of the minimum spanning tree for some function $g$. This
conjecture implies the {\em efficient} polynomial-time approximation scheme
(PTAS) of the traveling salesperson problem in $H$-minor free graphs; that is,
a PTAS whose running time is of the form $2^{f(\epsilon)}n^{O(1)}$ for some
function $f$. The state of the art PTAS for TSP in H-minor-free-graphs has
running time $n^{1/\epsilon^c}$. We take a further step toward proving this
conjecture by showing that if the bounded treewidth graphs have light greedy
spanners, then the conjecture is true. We also prove that the greedy spanner of
a bounded pathwidth graph is light and discuss the possibility of extending our
proof to bounded treewidth graphs.
