  For a connected graph $G$, the Wiener index, denoted by $W(G)$, is the sum of
the distance of all pairs of distinct vertices and the eccentricity, denoted by
$\varepsilon(G)$, is the sum of the eccentricity of individual vertices. In
\cite{Kc}, the authors posed a conjecture which states that given a graph $G$
with at least three vertices, the difference between $W(G)$ and
$\varepsilon(G)$ decreases when an edge is contracted and proved that the
conjecture is true when $e$ is a bridge. In this manuscript, we confirm that
the conjecture is true for any connected graph $G$ with at least three vertices
irrespective of the nature of the edge chosen.
