Batyrev's index boundedness conjecture for log terminal Fano varieties

Let $X$ be a normal $n$-dimensional log terminal Fano variety over an algebraically closed field $k$, with $K_X$ ${\mathbb Q}$-Cartier and $-K_X$ ample. Its index is the minimal positive integer $I$ such that $I\cdot K_X$ is Cartier. A set of varieties is bounded if its members occur as geometric fibers of a finite-type family over $k$.

Index boundedness conjecture. The family of $n$-dimensional log terminal Fano varieties of index $I$ is bounded.

The source attributes this weaker conjecture to V. Batyrev and notes that it follows from the BAB conjecture by taking $\epsilon=1/I$. Its resolution status is not established by the supplied evidence.