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

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