Characteristic-dependence conjecture for syzygies of Veronese embeddings

Let $r\geq 7$, and let $d$ be sufficiently large. Consider the $d$-uple embedding of $\mathbb P^r$ and its Betti table. Characteristic-dependence conjecture. For any $d\gg 0$, the Betti table of $\mathbb P^r$ under the $d$-uple embedding depends on the characteristic. This conjecture is motivated by characteristic-dependent torsion in syzygies of random flag complexes; for $d>2$, the source notes that no such torsion examples were known for Veronese embeddings.