The maximal-subfields algebraicity conjecture for division rings

Let $D$ be a division ring with center $F$. A subfield $K$ of $D$ is maximal if it is maximal among subfields of $D$. Maximal-subfields algebraicity conjecture. If every maximal subfield of $D$ is algebraic over $F$, then $D$ is algebraic over $F$. The source explains that the assertion would strengthen a question answered negatively when only one maximal subfield is assumed algebraic; it gives no resolution of the all-maximal-subfields version.