The cut-partition conjecture for t-triangles in fullerenes

Let $C_{56,622}$ and $C_{80,31924}$ denote the indicated fullerene graphs. A cut-partition of a hexagonal subgraph is the set of components generated by the stated cut-partition construction, and a $t$-triangle is the triangle type defined in the source. Cut-partition t-triangle conjecture. $C_{56,622}$ and $C_{80,31924}$ are the only fullerenes whose cut-partition consists solely of $t$-triangles.

The claim is based on numerical results about cut-partitions of hexagonal subgraphs. No proof or disproof is supplied in the source.