The globally F-regular type implies log Fano conjecture

Let $X$ be a projective variety of characteristic zero. A variety is globally F-regular type if it has globally F-regular reductions modulo a Zariski-dense set of primes, and it is log Fano if there is an effective $\mathbb Q$-divisor $\Delta$ such that $(X,\Delta)$ is Kawamata log terminal and $-(K_X+\Delta)$ is ample.

Globally F-regular type implies log Fano. A projective globally F-regular type variety of characteristic zero is log Fano.

The converse to the corresponding positive-characteristic implication is open in general. It is known under the additional hypothesis that the variety is a $\mathbb Q$-factorial Mori Dream space; if true in general, the conjecture would also imply that globally F-regular type varieties are Mori Dream spaces.