  We consider the flow of two viscous and incompressible fluids within a
bounded domain modeled by means of a two-phase Navier-Stokes system. The two
fluids are assumed to be immiscible, meaning that they are separated by an
interface. With respect to the motion of the interface, we consider pure
transport by the fluid flow. Along the boundary of the domain, a complete slip
boundary condition for the fluid velocities and a constant ninety degree
contact angle condition for the interface are assumed.
  The main result of the present work establishes in 2D a weak-strong
uniqueness result in terms of a varifold solution concept \`a la Abels
(Interfaces Free Bound. 9, 2007). The proof is based on a relative entropy
argument. More precisely, we extend ideas from the recent work of Fischer and
the first author (Arch. Ration. Mech. Anal. 236, 2020) to incorporate the
contact angle condition. To focus on the effects of the necessarily singular
geometry, we work for simplicity in the regime of same viscosities for the two
fluids.
