Brauer $p$-dimension bounds for henselian discretely valued fields

Let $K$ be a henselian discretely valued field with residue field $F$ of characteristic $p>0$. Assume that $[F:F^p]=p^n$.

Brauer $p$-dimension conjecture. Then

$$n\leq \operatorname{Br.dim}_p(K)\leq n+1.$$

This conjecture seeks a sharp upper bound for the Brauer $p$-dimension in residual characteristic $p$. The lower bound is known, while the asserted upper bound is the remaining open part.