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Higher-spin theory contains a complex coupling parameter $\eta$. Differenthigher-spin vertices are associated with different powers of $\eta$ and itscomplex conjugate $\bar \eta$. Using $Z$-dominance Lemma, that controlsspin-locality of the higher-spin equations, we show that the third-ordercontribution to the zero-form $B(Z;Y;K)$ admits a $Z$-dominated form that leadsto spin-local vertices in the $\eta^2$ and $\bar \eta^2$ sectors of thehigher-spin equations. These vertices include, in particular, the $\eta^2$ and$\bar \eta^2$ parts of the $\phi^4$ scalar field vertex.

Spin-Locality of $\eta^2$ and $\bar\eta^2$ Quartic Higher-Spin Vertices