Let $K$ be an arbitrary field, and let $K^s$ be a fixed separable closure of $K$. Let $F/K$ a be a finite Galois extension in $K^s$. Let $n>0$ be a natural number. Let $A$ be a central simple algebra over $F$ such that $A^{\otimes n}$ is split. By Brauer's theorem, $A$ can be split by a finite separable extension $L/F$ of degree dividing $n^r$ for some $r\ge 0$; see Theorem 2.8.7 in the book by Philippe Gille and Tamás Szamuely "Central simple algebras and Galois cohomology", 2nd edition, Cambridge 2017.
Conjecture. There exists a finite separable extension $C/K$ in $K^s$ of degree dividing $n^{r}$ for some $r\ge 0$ such that the extension $L=CF$ of $F$ splits $A$.
I can prove this conjecture in the case when $K$ is a local or global field.
Question. Is this conjecture true over an arbitrary field (or over any field of characteristic 0)?
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