There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section.
With property 9, we see that we can drop metacompactness from Traylor's theorem, but at the cost of a set-theoretic assumption.
Another example of this is Fleissner's theorem that the axiom of constructibility implies that locally compact, normal Moore spaces are metrizable.
Nyikos proved that, under the so-called PMEA (Product Measure Extension Axiom), which needs a large cardinal, all normal Moore spaces are metrizable.
Jones (1937) gave an example of a pseudonormal Moore space that is not metrizable, so the conjecture cannot be strengthened in this way.