  The extension of nonlinear higher-spin equations in d=4 proposed in
[arXiv:1504.07289] for the construction of invariant functional is shown to
respect local Lorentz symmetry. The equations are rewritten in a manifestly
Lorentz covariant form resulting from some Stueckelberg-like field
transformation. We also show that the two field-independent central terms
entering higher-spin equations which are not entirely fixed by the consistency
alone get fixed unambiguously by the requirement of Lorentz symmetry. One of
the important advantages of the proposed approach demonstrated in the paper is
the remarkable simplification of the perturbative analysis.
