I believe very strngly that it is false. But the question is: How do you prove/disprove it??
I think that it has been proven that it is impossible to either prove or disprove it using the Zermelo-Fraenkel axioms of set theory plus the axiom of choice.
Therefore it is independent of the axioms.
I think that it is analogous to the parallel postulate in geometry.
If you assume it is true, then you have Euclidean geometry. If you assume it is not true, then you have non-Euclidean geometry.
In Euclidean geometry, the angles of a triangle add to 180 degrees, but this is not true in non-Euclidean geometries.