Spin-Locality of $eta^2$ and $bareta^2$ 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.