How can you auto-calibrate a conductance sensor?
The instruments that measure bulk conductivity1 are calibrated with standard solution.  As solution with known conductivity is introduced and the cell constant is measured.  The conductivity cell and the circuitry are calibrated at the same time.

Such calibration would be done daily or weekly.  I wouldn't call it fully automatic calibration, though, because the standard solution is introduced by an operator.

As Hkob had mentioned in his comment, contamination and chemical reactions can gradually change the cell constant of the conductivity cell.  Those are faster and more pronounced that the drifts in the analog front end.  The calibration with standard solution takes care of both.

Additional reading
Conductivity Theory and Practice (PDF, p.20 deal with determination of the cell constant)

1 Conductivity of water is one of the indicators of quality, for example.