No, not necessarily. It's standard practice to pick a p-value significance cut-off (0.05), but report the smallest such standard cut-off that any particular value meets. So "p < 0.001" is reported for values that meet that threshold. Anything over the cut-off is just not reported as significant.

That seems dishonest to me. They're saying that some results are more significant after the fact. Is there any mathematical justification for why this is OK?

As far as I know there is not. There is no such thing as "more significant", results either are significant or they're not.

That being said, the authors are doing a lot of comparisons here! They ought to be reporting Dunn-Bonferroni corrected p-values.