We report a one-dimensional planar optical waveguide with a transverse distribution of the inhomogeneous loss profile, which exhibits an exceptional point (EP). The waveguide hosts two leaky resonant modes, where the interaction between them in the vicinity of the EP is controlled by proper adjustment of the inhomogeneity in the attenuation profile only. We study the adiabatic dynamics of propagation constants of the coupled modes by quasistatic encirclement of control parameters around the EP. Realizing such an encirclement with the inhomogeneous loss distribution along the direction of light propagation, we report the breakdown of adiabatic evolution of two coupled modes through the waveguide in the presence of an EP. During conversion the output mode is irrespective of the choice of the input excited mode but depends on the direction of light transportation. This topologically controlled, robust scheme of asymmetric mode conversion in the platform of the proposed all-lossy waveguide structure may provide a means for implementation of state-transfer applications in chirality-driven waveguide-based devices. © 2018 American Physical Society.