The DA's application to the Western Cape High Court to challenge the prohibition on the personal care industry, which includes hairdressers and beauticians, was postponed on Friday.
The party and co-applicant Palladium, a hairdressing business, contend in papers Dlamini-Zuma had failed to make it clear who would formulate the directions relating to how and when the industry could resume work, and which grounds would be used to decide when it could reopen, and by when these directions would be completed.
The DA and Palladium contend in their application many businesses have opened and even when it was Level 5, businesses that involved close contact with other people had operated.
The DA said around half of the people in the industry were considered "semi-skilled" and their income prevented poverty.
"The industry is simply prohibited from operating indefinitely, subject to the whims of some unidentified minister," submitted the DA in court papers.