We explore a local form of second-order Vasiliev equations proposed in [arXiv:1706.03718] and obtain an explicit expression for quadratic corrections to bosonic Fronsdal equations, generated by gauge-invariant higher-spin currents. Our analysis is performed for general phase factor, and for the case of parity-invariant theory we find the agreement with expressions for cubic vertices available in the literature. This provides an additional indication that local frame proposed in [arXiv:1706.03718] is the proper one.
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N. G. Misuna (2018) studied this question.
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