The conductivity $\kappa$ of an electrolyte solution does not depend on

Conductivity is an intensive property of the solution: $\kappa =G(l/A)$, so the geometry of the cell cancels out. It depends on the number of ions per unit volume (concentration), their mobility (which depends on temperature and solvent viscosity) and the nature of the ions. Only the measured conductance $G$, not $\kappa$, changes with electrode separation.