So I assume you are questioning how can A and B be correlated when A shows a significant difference between the groups while B does not? Significant here refers to statistical significance as opposed to saying that they differ largely. The former refers to whether there is statistical justification to make a claim, and the latter refers to the size of the difference.
While the authors results seem paradoxical, from a mathematical standpoint it isn't because these are statistical tests (i.e. inferences about the population are made based on samples) and you may not have large enough sample sizes to elimanate errors in the conclusions. In fact, the 0.05 threshold says that there is a 5% chance that you say they are correlated (or that they differ on the two groups) when they are NOT.
Don't get me wrong. These are inconsistent statements. What it shows is that the sample size(s) was not sufficiently large to resolve the questions. In other words, the authors got mixed results.
For smaller sample sizes, you have larger error bars, i.e. your sample statistics have more variance. So it becomes harder to justify that A & B are different on the two groups or that they are correlated (the rejection region becomes smaller). So it is possible to have the results from 2 of the tests contradict the result from the other test. It must be that for at least one of the tests an error was made.
I don't mean to say the authors erred. It just means that the statistical methods established (hypothesis testing) point you to a conclusion for each test, but those methods cannot remove all errors (saying yes when the answer is no, or vice versa) because you are sampling the population.
If you were to redo the tests with larger sample sizes then you should be able to eliminate the contradiction.