Herzog's Betti-number inequality for symmetric and exterior shifting

Let $\Gamma$ be a simplicial complex, let $\Delta(\Gamma)$ and $\Delta^e(\Gamma)$ denote its symmetric and exterior algebraic shifts, and let $I_{\Delta(\Gamma)}$ and $I_{\Delta^e(\Gamma)}$ be their Stanley–Reisner ideals. Herzog's conjecture. For every simplicial complex $\Gamma$,

$$\beta_{i,j}(I_{\Delta(\Gamma)})\leq \beta_{i,j}(I_{\Delta^e(\Gamma)}).$$

This conjecture concerns the relationship between general graded Betti numbers under symmetric and exterior algebraic shifting; the source gives no resolution status.