Largest-vanishing-range conjecture for quasi-smooth 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 a positive integer $n$, let $X$ be the $n$-fold from the quasi-smooth Calabi–Yau hypersurface construction in Proposition 2. Largest-vanishing-range conjecture. There is a largest possible positive integer $M$ such that

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

and this $X$ realizes that largest $M$ among all quasi-smooth Calabi–Yau hypersurfaces of dimension $n$. Low-dimensional computations support the claim, while its validity for arbitrary dimension remains open.