Herzog's minimality conjecture for exterior generic annihilator numbers

Let $E$ be the exterior algebra on $E_1$, let $J\subseteq E$ be a graded ideal, and let $v_1,\ldots,v_n$ be a basis of $E_1$. Write $\alpha_{i,j}(E/J)$ for the exterior generic annihilator numbers of $E/J$, and $\alpha_{i,j}(v_1,\ldots,v_n;E/J)$ for the exterior annihilator numbers with respect to this basis.

Herzog's minimality conjecture. For $1\leq i\leq n$ and $j\geq 0$,

$$\alpha_{i,j}(E/J)\leq \alpha_{i,j}(v_1,\ldots,v_n;E/J).$$

The conjecture asks whether the exterior generic annihilator numbers are minimal among the exterior annihilator numbers associated to all bases. The surrounding discussion states that the conjecture is false and provides a counterexample; corresponding symmetric generic annihilator numbers are also shown to admit a counterexample after slight modifications.