This is probably stupid but I don't see an argument.
Let \begin{eqnarray} P&\rightarrow & X\\ \downarrow&&\downarrow\\ Y&\rightarrow& Z \end{eqnarray} be a homotopy cartesian diagram of simplicial sets. You can assume for example that $X\to Z$ is a Kan fibration and the diagram is cartesian in the categorical sense.
Is \begin{eqnarray} \pi_0P&\rightarrow & \pi_0X\\ \downarrow&&\downarrow\\ \pi_0Y&\rightarrow& \pi_0Z \end{eqnarray} a cartesian diagram of sets?