Is the electric conductivity of chlorine ions influenced by its oxidation state?
It is the similar effect as for molar conductivity of cations of alkali metals, which grows from $\ce{Li+}$ toward $\ce{Cs+}$, in spite of increasing ionic radius.
The point is, the effective radius of the hydrated ions has the opposite trend as the &quot;naked&quot; ion radius trend. $\ce{Li+}$ ion with the smallest radius has the strongest electrostatic field and the largest hydration sphere and has therefore the lowest molar conductivity.
The similar effect I expect for chlorine oxyacid anions. In this case, there is also decreasing degree of delocalization of the negative charge in trend $\ce{ClO4-} \gt \ce{ClO3-} \gt \ce{ClO2-}$.