The iterated Betti-number inequality for symmetric and exterior shifting

Let $\Gamma$ be a simplicial complex, and let $b_{i,j}(\Gamma)$ and $b^e_{i,j}(\Gamma)$ denote its symmetric and exterior iterated Betti numbers, respectively. The iterated Betti-number conjecture. For every simplicial complex $\Gamma$,

$$b_{i,j}(\Gamma)\leq b^e_{i,j}(\Gamma).$$

The inequality is proposed in analogy with Herzog's conjecture on graded Betti numbers. The source records equality for sequentially Cohen–Macaulay complexes, but gives no resolution of the general case.