Lüroth-type uniqueness conjecture for invariant fields in infinitely many variables

Let $\Psi$ be an infinite set and let $F|k$ be a non-trivial regular field extension. Write $F_{\Psi}$ for the field obtained from $F$ in the permuted variables, and let $K|k$ be an $\mathfrak{S}_{\Psi}$-invariant field extension contained in $F_{\Psi}$. The claim concerns a field extension $L|k$ contained in $F$ satisfying the following conditions:

Lüroth-type conjecture. There is a unique field extension $L|k$ in $F$ such that

$$K\subseteq L_{\Psi},$$

$L$ is algebraically closed in $F$, and, for every field extension $L'|k$ contained in $L$ with $\operatorname{tr.deg}(L'|k)<\infty$, there is a field extension $L”|L'$ contained in $L$ such that

$$\operatorname{tr.deg}(L”_{\Psi}|K\cap L”_{\Psi})<\infty.$$

This is a Lüroth-type structural assertion for invariant subfields of fields of rational functions in infinitely many permuted variables. The supplied text does not indicate whether the assertion is known or remains open, so its status is recorded as open.