Optimal Brauer p-dimension conjecture for complete discretely valued fields

Let $\mathcal{K}$ be a complete discrete valued field of characteristic zero with residual characteristic $p$, and let $\kappa$ be its residue field. For a field $E$ of characteristic $p$, define its $p$-rank $\mathscr{R}_p(E)$ by

$$[E:E^p]=p^{\mathscr{R}_p(E)}.$$

The Brauer $p$-dimension $Br_pdim(F)$ is the least integer $\ell$ such that $ind(\alpha)\mid per(\alpha)^\ell$ for every finite extension $E/F$ and every $\alpha\in Br(E)[p^\infty]$. Optimal Brauer $p$-dimension conjecture.

$$\mathscr{R}_p(\kappa)\leq Br_pdim(\mathcal{K})\leq \mathscr{R}_p(\kappa)+1.$$

The lower bound is known, while the upper bound remains open. The conjecture predicts the optimal Brauer $p$-dimension in the mixed-characteristic complete discretely valued setting.