Zhang, Liu and Zhou's extremal Eccentric Connectivity Index conjecture

For integers $n$ and $m$, define

$$d = \left\lfloor \dfrac{2n+1-\sqrt{17+8(m-n)}}{2} \right\rfloor$$

and let $E_{n,m}$ be the graph obtained from a clique $K_{n-d-1}$ and a path

$$P_{d+1}=v_0v_1\ldots v_d$$

by joining each clique vertex to both $v_d$ and $v_{d-1}$, and by joining $m-n+1-\binom{n-d}{2}$ clique vertices to $v_{d-2}$. Write $d_{n,m}$ for the relevant parameter of this construction. Zhang, Liu and Zhou's conjecture. If $d_{n,m}\geq 3$, then $E_{n,m}$ is the unique graph with maximal eccentric connectivity index among all connected graphs with $n$ vertices and $m$ edges.

The conjecture proposes the extremal and unique structure for the Eccentric Connectivity Index in the specified range. PHOEG is presented as a tool for testing such claims computationally, but the supplied text does not state whether this conjecture has been proved or refuted.