**Question:**

given a complete symmetric graph $G(V,E)$ with $n$ vertices and edges $e_{ij}$ having weight $\omega_{ij}$, does there always exists a vector of vertex potentials $(\pi_1,\,\dots,\,\pi_n)$ that, when added to the weight of adjacent edges, guarantee that the minimum 1-tree of the Held-Karp lower bound resembles the edge-set of optimal tour, i.e. of the lightest Hamilton cycle of $G$.

I have scrutinized some related websites but couldn't find an *explicit* statement about the existence of such a vector, but only estimates,
so a clarification would be of help.