= Solution
Under inversion $\mathbf x\mapsto-\mathbf x$, the <spherical shell> and its volume element are unchanged, while the product of three <scalar products> changes sign. Its integral is therefore its own negative:
$$
\boxed{\int_V(\mathbf x\cdot\mathbf n_1)(\mathbf x\cdot\mathbf n_2)
(\mathbf x\cdot\mathbf n_3)\,d^3x=0}.
$$
Equivalently, its rank-three moment <tensor> is completely symmetric and cannot have a nonzero multiple of the antisymmetric <Levi-Civita symbol>.
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