Largest-vanishing-range conjecture for canonical Calabi–Yau hypersurfaces

Let $s_0=2$ and $s_m=s_{m-1}(s_{m-1}-1)+1$ for $m\geq 1$ be Sylvester's sequence. For an integer $n\geq 2$, set

$$d=(s_{n-1}-1)(3s_{n-1}-4)^2.$$

Let $X$ be a general hypersurface of degree $d$ in

$$\mathbb{P}^{n+1}(d/s_0,\ldots,d/s_{n-2},(s_{n-1}-1)(3s_{n-1}-4),(s_{n-1}-1)(3s_{n-1}-5),3s_{n-1}^2-9s_{n-1}+7).$$

It is a quasi-smooth canonical Calabi–Yau hypersurface of dimension $n$ with ample Weil divisor $\mathcal{O}_X(1)$. Largest-vanishing-range conjecture. Among all such hypersurfaces, $X$ has the largest integer $M$ such that

$$H^0(X,\mathcal{O}_X(\ell))=0\qquad\text{for }1\leq \ell<M,$$

and this integer is $M=3s_{n-1}^2-9s_{n-1}+7>2^{2^{n-1}}$. The examples attain the largest possible value in the low dimensions cited, but the general optimality assertion remains open.