Smallest minimal log discrepancy conjecture for exceptional Fano varieties

Let $s_n$ be the $n$th Sylvester number, and let $n\geq 2$ be an integer. For even $n$, the candidate value is

$$\frac{4(s_n-1)}{s_n^3-9s_n}.$$

For odd $n$, the candidate value is

$$\frac{4(s_n-3)}{s_n^3-19s_n+14}.$$

Smallest minimal log discrepancy conjecture. For any even (respectively, odd) integer $n\geq 2$, the smallest minimal log discrepancy of an exceptional Fano variety of dimension $n$ is

$$\frac{4(s_n-1)}{s_n^3-9s_n}\left(\text{respectively }\frac{4(s_n-3)}{s_n^3-19s_n+14}\right).$$

The paper constructs exceptional Fano hypersurfaces attaining these values in every dimension, and conjectures that they are optimal among all exceptional Fano varieties of the same dimension. The conjectured minimum is asymptotic to $4/s_n^2$, hence decreases doubly exponentially with $n$; the general optimal lower bound remains open.