Strength of "Every infinite set can be ordered in a dense order"

Question

Assuming $\mathsf{AC}$, it is easy to show that every infinite set can be ordered in a dense order: for a set of cardinality $\aleph_0$ we can take a bijection into $\Bbb{Q}$; and for a general infinite cardinal $\kappa$ we can take $\kappa$ copies of $\Bbb{Q}$ (with an infinite lexicographic order).

What is the strength of the statement "every infinite set can be ordered in a dense order"? Is it equivalent to $\mathsf{AC}$?

Answer 1

It is so weak that it won't even prove the Boolean Prime Ideal theorem. Indeed, even assuming $\sf ZF+BPI$, the principle "Every infinite set can be densely ordered" is not even enough to prove the Kinna–Wagner Principle.

Pincus, David, The dense linear ordering principle, J. Symb. Log. 62, No. 2, 438-456 (1997). ZBL0890.03022.