Here is a link to the full text, rather than the news story.
Time trends and factors related to COVID-19 vaccine hesitancy from January-May 2021 among US adults: Findings from a large-scale national survey
From the full text, I believe that the study was conducted using a Facebook survey to measure the change in hesitancy over time, in which respondents were asked several questions about their age, sex, race, education level, etc. and attitudes toward vaccinations.
It doesn't appear that the education level of PhD was one of the options. Instead, respondents could check a category called "Professional Degree", which is where the PhD would come in, along with MD, JD. etc but it's vague, so someone with certification in any profession might check this category. I dare say that the less education they have, the more likely they would be to misunderstand the category. So the reliability of this answer is in question.
It becomes much more suspect when you see the pattern of results shown in the other levels of education, and when you read other research articles which are consistently showing that the higher the level of education one has, the less likely one is vaccine hesitant. I linked just a few studies in a post above.
My nephew just completed a Masters Paper on this topic, using a Facebook survey very similar to the one in this article. His paper was rejected because the data could not be confirmed. Too many people use Facebook surveys to make statements, or give silly answers.
Even within the paper, the authors state, "we assume the survey was completed in good faith". But this is not likely to be true. Furthermore, if someone is lying on an anonymous Facebook survey, then wouldn't they claim to have the highest category of education? If any category is susceptible to fraud, it is the level of education.
For the results to be meaningful, there needs to be a better way of making sure the data is accurate. I will be surprised if this paper gets printed.
jmo