Maximal Wiener index conjecture for fullerene graphs

Let $G$ be a fullerene graph, and let $W(G)$ denote its Wiener index. A graph is nanotubical if it belongs to one of the nanotubical fullerene families with caps of types $a$--$d$ described in the preceding propositions.

Maximal Wiener index conjecture. If a fullerene graph of an arbitrary order has the maximal Wiener index, then it is a nanotubical fullerene graph with caps of types $a$--$d$, and its Wiener index is given by Propositions

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The conjecture proposes a complete description of the fullerene graphs attaining the maximal Wiener index, based on the computations for fullerene graphs with at most $216$ vertices. The cited propositions provide the formulas for the relevant nanotubical families; the assertion for arbitrary order remains open in the supplied source.