The lower-bound conjecture for the first gap of global log canonical thresholds

For every $n\geq 2$, let $m_n$ be the integer defined by the construction in Theorem. Let $(X,(1-b)S)$ be a complex klt Calabi–Yau pair of dimension $n$, where $S$ is a nonzero effective Weil divisor. First-gap conjecture. One has

$$b\geq \frac{1}{m_n}.$$

The paper constructs examples with $b=1/m_n$, so the conjecture would identify the first gap in this setting and establish optimality of the construction. It remains open.