Axiom of Choice and bases for vector spaces over a specific field

• Blass proved in $\mathsf{ZF}$ that if every vector space has a basis then the Axiom of Choice holds [Bla83].
• Keremedis showed in $\mathsf{ZFA}$ that if, in every vector space over $\mathbb{F}_2$, every generating set contains a basis then the Axiom of Choice holds [Ker96].
• Keremedis showed in $\mathsf{ZFA}$ that if every vector space over $\mathbb{Q}$ has a basis then for each $n$, every well-ordered family of $n$-element sets has an infinite subset with a choice function [Ker01].
• Morillon showed over $\mathsf{ZFA}$ that if every vector space over $\mathbb{F}_2$ has a basis then the Axiom of Choice for families of sets of size $2$ holds [Mor09].

There is a related question stemming from Blass's theorem. Blass showed in $\mathsf{ZF}$ that if every vector space has a basis then the Axiom of Choice holds. However, his proof does not work over $\mathsf{ZFA}$, i.e. $\mathsf{ZF}$ with the Axiom of Extensionality modified to allow atoms. It is an open question whether the implication holds over $\mathsf{ZFA}$.