TDS = K (conductivity or 1/resistivity) under normal conditions. K is the factor he is referring. What we do in conductivity meter calibration is entering the value of K by using a standard.
Your TDS measurments are valid only in the range of pH= 6-8. Because at lower pH protons which has highest mobility gives the highest contribution to the conductivity than the contribution form TDS.
At high pH OH- which has relatively higher mobility gives the highest contribution to the conductivity than the contribution form TDS.
Once you analyzed cations and anions from other analysis (Eg AAS, Ion selective electrodes) that give more accurate answer. Once you compare those results if you are getting a higher results that is due the unsuitable pH value.