A D-moduli problem?

We point out a generic problem in string-inspired supergravity models with an anomalous $U(1)_X$. A large number of matter multiplets charged under $U(1)_X$ remain massless above the supersymmetry-breaking scale because of degeneracy of vacua solving the D-flatness conditions. A toy model is analyzed as an illustration of the mechanism; we find the surprising result that many scalars remain massless after supersymmetry-breaking in a hidden sector.