The Diophantine-dimension conjecture for effective index

Let $F$ be a field, let $\alpha\in H^n(F,\mu_\ell^{\otimes n})$, and suppose that the Diophantine dimension of $F$ is bounded by $d$. Write $\operatorname{eind}(\alpha)$ for the effective index of $\alpha$.

Effective-index conjecture. One should have

$$\operatorname{eind}(\alpha)\mid \ell^{\binom{d-1}{n-1}}.$$

The conjecture proposes an optimistic improvement to the bounds obtained earlier in the paper. It arose from discussions at the AIM conference on deformation theory, patching, quadratic forms, and the Brauer group; no resolution is given in the supplied text.