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When a fluid or Ising-like magnet is confined between two parallel walls that are each completely wet by different bulk phases, the interface separating the phases is subject to large-scale fluctuations determined by the slit width. It was noted some time ago that, in two dimensions, the scaling expression for the probability distribution function describing the interfacial height across the slit shows remarkable similarities with predictions of conformal invariance. However, this local scale invariance appears to contradict the strongly anisotropic nature of (1 + 1)-dimensional interfacial fluctuations along and perpendicular to the interface, characterized by the wandering exponent. In this paper, we show that similarity with conformal invariance is not coincidental and can be understood explicitly as the projection of a local scale invariance for a wandering line in 2 + 1 dimensions.