The cohomological criterion for exceptional syzygy bundles to be Steiner

Let $n$ be odd and set $i=\frac{n+1}{2}$. Let $F_{i-1}$ and $F_i$ be consecutive syzygy bundles in the sequence of bundles defined by the short exact sequences

$$0\longrightarrow F_{i-1}\longrightarrow \mathcal O_{\mathbb P^n}^{\beta_i}(d_i-d_{n+1})\longrightarrow F_i\longrightarrow 0.$$

The relevant cohomology groups are

$$H^{n-i+1}\bigl(F_{i-1}(d_{n+1}-d_i)\bigr)\quad\text{and}\quad H^i\bigl(F_i^\lor(d_i-d_{n+1})\bigr).$$

Cohomological criterion for syzygy bundles. The two cohomology groups are isomorphic if and only if they are both zero; consequently, the syzygy bundles $F_i$ are Steiner if and only if they are exceptional.

The proposed criterion would establish the converse of the preceding result, which shows that under the stated hypotheses Steiner syzygy bundles are exceptional. The source presents this as a desired result rather than a proved theorem, so its resolution is not established here.