The minimum eccentricity-entropy conjecture for graphs

Let $G$ be a graph of order $n$, and let $I_{ecc}(G)$ denote its eccentricity-entropy. An extremal graph is one attaining the minimum value of this entropy. The minimum eccentricity-entropy conjecture. Among graphs of order $n$, the minimum value of $I_{ecc}$ is attained by the graph obtained by removing a small number of edges from the complete graph of order $n$. In particular, extremal graphs of order $n$ have $k\geq \frac{n}{2}$ vertices of degree $n-1$. The paper states that this conjecture is true except for the assertion concerning the removal of a small number of edges.