Quantum hall transformer in a quantum point contact over the full range of transmission

A recent experiment [Cohen et al., Science 382, 542 (2023)] observed a robustly quantized $e^2/2h$ conductance in a quantum point contact between fractional quantum Hall edges (a quantum Hall transformer), which then vanishes at low temperature. While this behavior can be described by a boundary sine gordon (BSG) model derived from electron tunneling, the microscopic motivation for such strong tunneling is unclear. We use an alternative model based on sliding charge density waves and a density inhomogeneity to clarify the BSG description and calculate the conductance over the full range of transmission. Using a perturbative method and a matrix product state calculation, we draw a quantitative connection between the BSG model and the physical quantum Hall system.