Minimal anticanonical-volume terminal Fano conjecture

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

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

A general hypersurface $X$ of degree $d$ in

$$\mathbb{P}^{n+1}(d/s_0,\ldots,d/s_{n-2},s_{n-1}-1,s_{n-1}-2,1)$$

is a terminal Fano $n$-fold. Minimal anticanonical-volume terminal Fano conjecture. This $X$ has minimal anticanonical volume among terminal Fano $n$-folds. The claim is supported by the minimality results cited for dimensions $2$, $3$, and $4$, while the general statement remains open.