Krashen's period-index conjecture from diophantine dimension

Let $F$ be a field, let $d$ be an integer, and let $\ell$ be the prime appearing in the cohomology group. The diophantine dimension $\operatorname{dd}(F)$ is the smallest integer $r\geq 0$ such that every degree-$d'$ homogeneous form in more than $d'^{\,r}$ variables has a nontrivial zero over $F$, for every $d'\geq 1$, when such an integer exists. For $\xi\in H^d(F,\mu_\ell^{\otimes d})$, write $\operatorname{ind}(\xi)$ for its index. Krashen's conjecture. If $\operatorname{dd}(F)\leq d$, then for every $\xi\in H^d(F,\mu_\ell^{\otimes d})$, the index $\operatorname{ind}(\xi)$ divides $\ell$. The paper presents the result for semiglobal fields as an analogue under a cohomological-dimension hypothesis replacing $\operatorname{dd}(F)\leq d$; the conjecture itself is not resolved here.