Erman–Martinova sharpness conjecture for Veronese Betti bounds

Let $M$ be the coordinate ring of the Veronese, under the hypotheses of Theorem, and let $\beta_{i,i+1}$ denote the corresponding graded Betti numbers. Erman–Martinova conjecture. The bounds presented in Theorem are sharp. More specifically, $\beta_{i,i+1}=0$ for $i$ outside the ranges in that theorem. The conjecture asserts that the theorem's predicted ranges contain all nonzero entries in the relevant Betti rows; the supplied text gives computational evidence for small examples but does not state a resolution.