The character conjecture for distinguishing dual fullerene graphs

For a feasible $n$, let $T_n$ be a dual fullerene graph, with adjacency matrix $A$ and degree matrix $D$. For $lpha,\beta\rtf]in (0,1)$, define its $(lpha,\beta\rtf])$-character by

$$ch_{lpha,\beta\rtf]}(T_n)=\operatorname{tr}\left(\exp\left(lpha A+\beta D\rtf]\right)\right).$$

Character conjecture. For $lpha,\beta\rtf]\in(0,1)$ and feasible $n_1,n_2$, if $T_{n_1}$ and $T_{n_2}$ are dual fullerenes, then

$$T_{n_1}\simeq T_{n_2}\Longleftrightarrow ch_{lpha,\beta\rtf]}(T_{n_1})=ch_{lpha,\beta\rtf]}(T_{n_2}).$$

The claim asserts that one character value distinguishes dual fullerene graphs up to isomorphism for every parameter pair in the stated range. It is supported by numerical experiments in the source, which states no proof or counterexample.