Fujita–Odaka's delta-invariant criterion for K-stability

Let $X$ be a $\mathbb{Q}$-Fano variety, meaning that $X$ is a normal projective variety with klt singularities and $-K_X$ is $\mathbb{Q}$-Cartier and ample. The delta-invariant $\delta(X)$ is defined from log canonical thresholds of anticanonical divisors of $m$-basis type.

Fujita–Odaka's conjecture. $X$ is K-stable, respectively K-semistable, with respect to $-K_X$ if and only if

$$\delta(X)>1,\qquad\text{respectively}\qquad \delta(X)\geq 1.$$

Fujita and Odaka proved the sufficiency of these inequalities; the converse is the conjectural part, and the source notes that it would follow if Berman–Gibbs stability were equivalent to K-stability.