For example, some of the first names are "Lashonda", "Lakeisha", "Takwanda" in the "Black" list, whereas the "white" list includes names like "Geoffrey", "Kristen", "Jennifer". If the perceived social status of the names isn't matched, that could invalidate their results.
As a follow-up, it would be interesting to do a survey on the perceived social status of each of the names they used to see how that might have influenced the results.
What I'm saying is that even if that were true, unless you're saying only "objectively" Black and White people are to judge the social status of Black and White names (such that each race judges their own name) - it would end up being no different than the exercise in the paper.
For example:
Let's say Denise is an "high status" Black name and LeShawn is a "low status" Black name. "Billy" is low status White and "Alexander" is high status.
Even if you knew this you'd need to know how the other race perceives the name since that's ultimately what's being judged here, hence it would be moot.
And that could be done through research surveys to account for that variable in this research.
Perhaps if Black people and White people had a "pure" view of other Blacks and Whites, unaffected by history you could do what you're saying.
Another example, to split racism from classism but with the confounding variable of national origin: "Lashonda Brown" vs "Abena Nwachukwu" (this is an actual last name that I Googled). On the resume, both could be given membership in a "Women's Society of Some University" to indicate gender, and same for other characteristics.
Edit: Note that even the SPELLING of "Geoffrey" vs "Jeffrey" vs "Jeff" may indicate something about social class or background.
Random source, I didn't keep all my tabs open: https://www.thedailybeast.com/the-elites-top-50-baby-names
In other words, you're trying to separate racism from classism, but race and class are already inexorably linked to begin with. Any perception of class to the name would already be influenced by historical racism, for better or for worse.