In the paper "Łoś's theorem and the boolean prime ideal theorem imply axiom of choice" Howard has shown that Łoś's theorem and the boolean prime ideal theorem imply axiom of choice. At the end of the paper, author remarks that it was not known at the time whether BPIT can be removed from their argument, but he mentions that if there are no free ultrafilters on any set, then Łoś's theorem (pretty trivially) holds, while choice doesn't, so we might be unable to remove BPIT.

Indeed, two years later Blass has shown that it is consistent with ZF that there are no free ultrafilters on any set, and hence Łoś's theorem doesn't imply axiom of choice.

This argument shows that in order to deduce full axiom of choice from Łoś's theorem, we need some ultrafilters. Here comes my question:

Is it known whether Łoś's theorem implies full axiom of choice in conjuntion with:

 

There exists a free ultrafilter

 

or

 

There exists a free ultrafilter on every infinite set

Thanks in advance

In the paper "Łoś's theorem and the boolean prime ideal theorem imply axiom of choice" Howard has shown that Łoś's theorem and the boolean prime ideal theorem imply axiom of choice. At the end of the paper, author remarks that it was not known at the time whether BPIT can be removed from their argument, but he mentions that if there are no free ultrafilters on any set, then Łoś's theorem (pretty trivially) holds, while choice doesn't, so we might be unable to remove BPIT.

Indeed, two years later Blass has shown that it is consistent with ZF that there are no free ultrafilters on any set, and hence Łoś's theorem doesn't imply axiom of choice.

This argument shows that in order to deduce full axiom of choice from Łoś's theorem, we need some ultrafilters. Here comes my question:

Is it known whether Łoś's theorem implies full axiom of choice in conjuntion with:

There exists a free ultrafilter

or

There exists a free ultrafilter on every infinite set

Thanks in advance