Lorentz Covariance of the $4d$ Nonlinear Higher-Spin Equations via BRST

We propose a BRST extension of the higher-spin gauge theory in $AdS_4$ with the BRST operator associated with the local Lorentz symmetry. Our construction supports manifest local Lorentz covariance and is applicable both to any homotopy scheme of the perturbative analysis including the recently proposed differential homotopy and to the variety of further extended higher-spin models in $AdS_3$ and $AdS_4$ with higher differential forms and Coxeter higher-spin algebras.