Bilinear Fronsdal currents in the $AdS_{4}$ higher-spin theory

We analyse higher-spin theory with general coupling constant $\eta$ at the second order, focusing on the gauge non-invariant vertices $\Upsilon(\omega,\omega)$, $\Upsilon(\Omega,\omega,C)$ and $\Upsilon(\omega,C)$, that are shown to generate nontrivial currents in the Fronsdal equations. Explicit expressions for the currents are found in the frame-like formalism counterpart of the TT gauge worked out in the paper. The nonlinear higher-spin theory is shown to generate all types of Metsaev's currents with the coupling constants manifestly expressed via the complex coupling constant $\eta$ of the higher-spin theory. It is shown that all currents in the higher-spin theory are conformal in the TT gauge except for those bilinear in the higher-spin gauge fields $\omega$.