Would an aqueous solution of Zn(2+) develop a positive charge if solid zinc was dipped into it?

Say you have a beaker containing $\pu{1M}$ solution of $\ce{ZnSO4}$. The solution will be electrically neutral because there will be an equal number of cations and anions. If you then put a zinc electrode into the solution, will some of the solid zinc reduce to give the solution a slightly positive net charge(because of the excess zinc ions)? Or will there be an equilibrium between the oxidation and reduction of zinc that would keep the solution electrically neutral?

Will the answer be any different in the case of copper-zinc galvanic cell with no salt-bridge? Will the electrolytic solution in the zinc half-cell of a copper-zinc galvanic cell with no salt-bridge develop a positive charge? The scenario is the same as the one in the previous passage except now the zinc electrode will be connected to a copper electrode.

For some context, I was trying to understand why the electrolytic solution in a galvanic cell needs to be electrically neutral in order for the electrodes to develop a non-zero potential difference.