  Signatures of quantum transport are expected to quickly vanish as dissipation
is introduced in a system. This dissipation can take several forms, including
that of particle loss, which has the consequence that the total probability
current is not conserved. Here, we study the effect of such losses at a quantum
point contact (QPC) for ultracold atoms. Experimentally, dissipation is
provided by a near-resonant optical tweezer whose power and detuning control
the loss rates for the different internal atomic states as well as their
effective Zeeman shifts. We theoretically model this situation by including
losses in the Landauer-B\"uttiker formalism over a wide range of dissipative
rates. We find good agreement between our measurements and our model, both
featuring robust conductance plateaus. Finally, we are able to map out the
atomic density by varying the position of the near-resonant tweezer inside the
QPC, realizing a dissipative scanning gate microscope for cold atoms.
