= Solution
Take canonical charged <chiral superfield> kinetic terms, gauge coupling $e>0$, and no <Fayet–Iliopoulos term>, since none is specified. Elimination of the three complex <auxiliary fields> gives
$$
F_+^*=-\lambda\varphi_0\varphi_-,\qquad
F_0^*=-\lambda\varphi_+\varphi_-,\qquad
F_-^*=-\lambda\varphi_+\varphi_0.
$$
The Abelian gauge <auxiliary field> obeys $D=-e(|\varphi_+|^2-|\varphi_-|^2)$ in this normalization. Thus the <F-term scalar potential> and the gauge <D-term> give
$$
\boxed{V=|\lambda|^2\left(|\varphi_0|^2|\varphi_-|^2
+|\varphi_+|^2|\varphi_-|^2+|\varphi_+|^2|\varphi_0|^2\right)
+\frac{e^2}{2}\left(|\varphi_+|^2-|\varphi_-|^2\right)^2.}
$$
All terms are nonnegative. The normalization of $e$ can be changed together with the vector-field normalization, but the relative charges and the zero-potential conditions cannot. A noncanonical <Kähler potential> would change the inverse-metric factors in the F-term potential; adding a <Fayet–Iliopoulos term> would shift $D$ and define a different model. Neither is silently introduced here.
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