This study explores an interesting aspect of eggshell morphology of parasitic bird species. It investigates properties of eggshells that could be associated with the cost of egg production (content of eggshell calcium) and with antimicrobial functions inside the nest (wettability and surface roughness). The methods and analyses are thorough and the manuscript is well written.
I found one of the hypotheses not to be very clear and I have some reservations about the use of surface roughness as a proxy for antimicrobial defense. Please find my specific concerns below.

Fitness reduction due to high risk of microbial contamination compared to egg damage by the host seems negligible, also considering the embryos of the host are also experiencing similar risk of contamination. Can the authors comment on this?

I was a little confused by the some of the hypotheses their predictions. For example, (Lines 125-127), the hypothesis that the eggs of brood-parasitic species exhibit greater roughness than non-parasitic species to decrease wettability (also line 180) implies that the parasitic eggs experience an increased risk of contamination compared to eggs of their hosts or eggs of non parasitic species. But this is idea is not supported by the background provided in L95. In any case, we would expect that both the parasitic eggs and eggs of their hosts should have higher surface roughness relative to non-parasitic and non-parasitized species. Could you clarify?

I think the surface roughness argument is at the moment the weakest point of the study. In my humble opinion, the authors offer a very simplistic and incomplete explanation for the link between roughness and its predicted function. The effect of surface microtopography on the settlement of microorganisms and the formation of biofilms will largely depend on the scale of the topographic features (and also the chemistry of the surface). Bacterial attachment will be greatest in valleys that are of similar dimension to the body size of the bacteria or on surfaces that have a large number of attachment points.
Could the authors expand a little about the scales at which eggshell roughness is expected to have anti attachment properties and whether the instrument they used evaluates roughness at these relevant scales?
Similarly, please consider that surface roughness can actually promote microorganism attachment (Crawford et al. 2012*). In this case we would expect that, the rougher a surface, the more likely it will be fouled with biofilms. Please integrate these ideas in the manuscript.

Also related to surface roughness, is it known what the effect of pigment deposition to eggshells (i.e. maculation) is on its surface roughness? Is it possible that the need for mimicry of egg maculation by parasites could result in similar eggshell roughness? Similarly, are there any observations of the cuticles of the eggs studied? Could these also contribute to roughness? These observations could help make a stronger case in favor of the “shared environment” hypothesis.

Cited: Crawford, R. J., Webb, H. K., Truong, V. K., Hasan, J., & Ivanova, E. P. (2012). Surface topographical factors influencing bacterial attachment. Advances in colloid and interface science, 179, 142-149.