Symbol-length conjecture for differential forms

Let $F$ be a field of characteristic $p>0$ and $[F:F^p]=p^n$ for $n\in \mathbb{N}_{>0}$. Let $\Omega^1_F$ be the module of absolute Kähler differentials and let $Z^1_F$ be its subgroup of closed forms. Assume one of the following:

• $F$ does not admit any finite extension of degree prime to $p$;
• $p=2$.

Symbol-length conjecture for differential forms. Then

$$\operatorname{len}(\Omega^1_F/Z^1_F)\leq n-1.$$

The conjecture refines the general bound supplied by the $p$-rank and is motivated by the vanishing of $\Omega^1_F/Z^1_F$ in $p$-rank one. Its general validity remains open.