The positive-characteristic decomposition conjecture for W-topologies

Let $K$ be a perfect field of positive characteristic, and let $\tau$ be a $W_n$-topology on $K$. Positive-characteristic decomposition conjecture. The topology $\tau$ is generated by $n$ independent V-topologies. This would imply the positive-characteristic case of the valuation-type conjecture, which says that the canonical topology on an unstable dp-finite field is a V-topology. The source notes that the claim is false in characteristic $0$, may hold in positive characteristic, and is known for $W_2$-topologies in odd characteristic.