Wiener complexity and diameter conjecture for maximal-Wiener fullerene graphs

Let $G$ be a fullerene graph, and let $W(G)$ denote its Wiener index. Let $C_W(G)$ denote its Wiener complexity and let $\operatorname{diam}(G)$ denote its diameter. Consider fullerene graphs of arbitrary order having maximal Wiener index.

Wiener complexity and diameter conjecture. The Wiener complexity and the diameter of fullerene graphs of an arbitrary order having the maximal Wiener index are given in Propositions

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Together with the preceding maximal-Wiener-index claim, this conjecture aims to determine further distance-based invariants of all extremal fullerene graphs. Its validity for arbitrary order is not established by the supplied computational discussion and remains open.