Understanding strong acid titrated with weak base graph
You are missing an important point. Conductometric titration is not just plotting the conductance after adding the titrant. One cannot plot conductance as it is, because you are significantly diluting the solution.  You have to apply a correction to take dilution into account. Once you apply a correction, the curve will remain flat because further addition of ammonia will not enhance conductivity.
See for example,

Conductometric titrations
Throughout a titration the volume of the solution is always increasing, unless the
conductance is corrected for this effect, non linear titration curves result. The
correction can be accomplished by multiplying the observed conductance either
by total volume (V+V') or by the factor (V+ V')/V, where V is the initial volume
of solution and V' is the total volume of the reagent added. The correction
presupposes that the conductivity is a linear function of dilution, this is true only
to a first approximation.