Why must this capacitor be a "low leakage" type?
Any leakage in C2 would only be corrected by the output of the opamp moving positive to supply the leakage current through R2.

If this moved the output close to the positive supply rail it would reduce the output swing capability or even saturating the opamp.  The actual effect will depend upon the value of R2.

The caution seems to be somewhat exaggerated.  In my experience it is rare for leakage in that capacitor to be an issue with the normal values of R1 and R2.

To avoid issues with stray capacitance R1 and R2 would usually no the higher than some 10's or 100's of kilohms. It would take many microamps of leakage in C2 for there to be a problem.

In most circuits it would be acceptable for C2 to be electrolytic or ceramic depending upon the value required.

If the circuit uses very high values of resistance and/or low supply voltages there could be more of an issue.