Batyrev's boundedness conjecture for the string-theoretic index

Let $X$ be an $r$-dimensional normal complex variety having at most Gorenstein canonical singularities. Its string-theoretic index is the least positive integer $l$ such that $e_{\mathrm{str}}(X)\in \frac{1}{l}\mathbb{Z}$. Batyrev's boundedness conjecture. The index $\operatorname{ind}_{\mathrm{str}}(X)$ is bounded by a constant $C(r)$ depending only on $r$. This conjecture asks for dimension-dependent uniform control of the denominators of string-theoretic Euler numbers; the supplied source does not state a resolution.