= Solution
Every global minimum found for nonzero $\lambda$ has $F_+=F_0=F_-=D=0$. By <energy positivity in global supersymmetry>, its zero energy means all <supercharges> annihilate it. Therefore
$$
\boxed{\text{Supersymmetry is unbroken throughout the vacuum family}.}
$$
The arbitrary neutral expectation $v$ is a supersymmetric modulus; the nonzero derivatives that would break <supersymmetry> vanish even though $v$ itself need not vanish. In the exceptional $\lambda=0$ case, every <D-flatness> minimum likewise has zero F- and D-auxiliaries, so <supersymmetry> remains unbroken even on the gauge-Higgsed branch. The <supersymmetric Higgs mechanism> permits internal gauge breaking without supersymmetry breaking. No extra constant or linear superpotential term is needed to find a zero-energy vacuum in this model.
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