The bottom theorem for separable cofinite subfields of random extensions

Let $K$ be a Hilbertian field and let $e\geq 1$ be an integer. For an $e$-tuple $\boldsymbol{\sigma}\in\operatorname{Gal}(K)^e$, write $K_s(\boldsymbol{\sigma})$ for the fixed field of the closed subgroup generated by its coordinates.

Bottom theorem. For almost all $\boldsymbol{\sigma}\in\operatorname{Gal}(K)^e$ and every proper subfield $N\subsetneqq K_s(\boldsymbol{\sigma})$, if the extension $K_s(\boldsymbol{\sigma})/N$ is separable, then it is infinite.

This is the separable modification of the bottom theorem problem for Hilbertian fields in positive characteristic, where purely inseparable finite extensions can occur. The supplied text does not state whether the assertion has been proved or remains open.