The étale index equals the index over iterated Laurent series fields

Let $k=\mathbb{C}((t_1))\cdots((t_d))$ be an iterated Laurent series field over the complex numbers, and let $\alpha\in\operatorname{Br}(k)$. Here $eti(\alpha)$ denotes the étale index and $ind(\alpha)$ the index of the Brauer class.

Étale-index conjecture.

$$eti(\alpha)=ind(\alpha).$$

The conjecture concerns equality of the étale index and the index over $d$-local fields of this form. The source gives supporting evidence from a result of Becher and Hoffman bounding the index by a power of the period, but does not state that the conjecture is resolved.