Yes. Even taking the Kentucky study at face value, the estimate odds ratio (probably similar to relative risk, given reinfection is fairly unlikely) is 2.34 (confidence interval 1.58 to 3.47), not "over 5-fold higher chances".

However, as they say, "...persons who have been vaccinated are possibly less likely to get tested. Therefore, the association of reinfection and lack of vaccination might be overestimated."