The separable bottom theorem for random extensions of Hilbertian fields

Let $K$ be a Hilbertian field and let $e\geq 1$ be an integer. For almost all \text{\boldmath\sigma}\in\operatorname{Gal}(K)^e, the field K_s(\text{\boldmath\sigma}) is a finite separable extension of no proper subfield.

Bottom theorem conjecture. For almost all \text{\boldmath\sigma}\in\operatorname{Gal}(K)^e, the field K_s(\text{\boldmath\sigma}) is a finite separable extension of no proper subfield.

This is the separability-modified form of the bottom theorem problem, introduced because for $K=\mathbb{F}_p(t)$ every separable extension has a proper cofinite subfield given by its $p$th powers. The supplied text does not state whether this modified problem has been resolved.