Answer
Yes, the solution will conduct electricity.
Work Step by Step
The complete ionic equation is the following:
$$Pb(NO_3)_{2(s}+2KI_{(s)} \Rightarrow PbI_{2(s)}+2K^{+}_{(aq)}+2NO_{3(aq)}^{-}$$
Because $KNO_3$ is soluble and dissociates into ions, the solution above the yellow precipitate is conductive.
