 Solution

For nonzero $\lambda$, zero potential requires both F-flatness and D-flatness:The product condition and equality of charged magnitudes together force ${\phi }_{+}={\phi }_{-}=0$. There is no condition on the neutral scalar. Consequently the global minima formOnly the neutral field can acquire a vacuum expectation value. It does not give the gauge vector a mass: its scalar kinetic term has no charged covariant derivative. Hence the gauged $U(1)$ remains unbroken in every global minimum for $\lambda \ne 0$. This is a neutral flat direction with oppositely charged chiral fields; a continuous vacuum family is not automatically gauge-symmetry breaking.

The exceptional uncoupled case $\lambda =0$ should be separated. Then only D-flatness remains, allowing $|{\phi }_{+}|=|{\phi }_{-}|=r$ and arbitrary ${\phi }_0$. For $r>0$, the charged expectations Higgs the $U(1)$; for $r=0$ it remains unbroken. The usual interacting answer assumes $\lambda \ne 0$.