Private sanctum would not prevent rope trick from being cast, but unless the rope trick extends beyond the ceiling of the private sanctum, rope trick would fail to permit entrance to the extradimensional space.
Rope trick says:
At the upper end of the rope, an invisible entrance opens to an extradimensional space that lasts until the spell ends.
Rope trick creates what the PHB calls a demiplane1:
Demiplanes are small extradimensional spaces with their own unique rules. They are pieces of reality that don’t seem to fit anywhere else. Demiplanes come into being by a variety of means. Some are created by spells, such as demiplane, or generated at the desire of a powerful deity or other force.
A demiplane is still a plane, and travelling to a demiplane is still planar travel, even if that travel takes the form of climbing up a rope into the demiplane.
To be clear, the prefix demi- means "inferior", so a demiplane is a particular type of plane, so "planar travel" includes "demiplanar travel".
1See this answer for a more detailed explanation the identification made between demiplane and extradimensional space.