Are employers allowed under German law to ask about vaccination status of their employees?

The pandemic situation has created a number of new obligations and duties of care for companies in Germany. For example, the most recent version of the SARS-CoV-2 Occupational Health and Safety Ordinance, which came into force in January 2021, requires companies to take measures to reduce contact in the workplace, to implement company hygiene concepts, to provide and wear mouth-nose protection masks and, in particular, to offer a free COVID-19 test at least twice per calendar week to employees who do not work exclusively at home (§§ 2 to 4 Corona-ArbSchV). Many of these measures are based on the vaccination status of employees, meaning companies could allow a lot of relief orders for vaccinated and recovered employees if they only knew which employees are affected. To better protect employees and customers from the higher infection burden of unvaccinated employees, they would like to ask about the vaccination status of their employees.