Ah, okay, so this route does not belong to Banedanmark, but to Hovedstadens Lokalbaner.

However, the web site of HL says:

Inden fusionen havde Hovedstadens Udviklingsråd overtaget statens aktier i privatbanerne, således at Hovedstadens Udviklingsråd ejer ca. 45% af aktierne i det nye selskab. Den øvrige aktiekapital ejes af amter og kommuner, som tidligere var aktionærer i de private baner.

^^ This means that all of HL belongs to government and municipalities. It also says that originally the routes had been largely privatized, but they were deprivatized in 2002.