The direct-sum extremality conjecture for matroids

Let $\mathsf{M}_1,\ldots,\mathsf{M}_s$ be matroids, and let $\mathsf{M}_1\oplus\cdots\oplus\mathsf{M}_s$ denote their direct sum. A matroid is extremal if its associated point is an extremal point of the matroid polytope under consideration.

Direct-sum extremality conjecture. If $\mathsf{M}_1,\ldots,\mathsf{M}_s$ are extremal matroids, then $\mathsf{M}_1\oplus\cdots\oplus\mathsf{M}_s$ is extremal.

The preceding theorem proves the converse implication: an extremal direct sum has extremal direct summands. The source describes this converse as subtle, and no resolution is supplied here.