In curious how can they make a statement such as “the study cannot be used to draw any conclusions” without providing any kind of reasoning, and get published in Nature?
The rebuttal was also not published in Nature, but it was closer, published in Nature Reviews Urology.
I think the main basis for the criticism is that they didn't look at clinical outcomes like infertility or ability to conceive. As a result, we only have a (very robust) report of a relationship between ibuprofen and hormone levels, not between ibuprofen and x, where x is some directly important/meaningful clinical outcome.
Drug companies hate it when a cash-cow drug is threatened in any way. Anything that threatens those profits is a potential target in one way or another. Researchers studying these drugs know just how much power and influence these drug companies have, and so, are afraid of angering them.
The reasoning is simple ~ make a vaguely disqualifying statement that hopefully keeps them in the good books.
Understanding people's incentives will often result in reasons to be suspicious, but suspicions aren't evidence and it's not a shortcut for understanding the science.