On consistency of the interacting (anti)holomorphic higher-spin sector

In the recently proposed generating systems for the (anti)holomorphic sector of the 4d higher spin theory and for the off-shell higher spin theory in generic dimension locality was achieved due to a peculiar limiting star product. Even though the generating systems exhibit all-order locality, the product itself encounters uncertainties when functions from specific classes are multiplied. This fact leads to the absence of the Leibniz rule for the differential operator acting on the auxiliary variables $z$ and, hence, its ambiguous definition in the generating equations. We identify the gap in the original proof of consistency associated with this freedom. Nonetheless considered generating systems are perfectly consistent as shown by direct computations on the resulting vertices. Considering specific orderings of fields we show that consistency rests on the star-exchange-like identities for the limiting star product formulated and proved here. Connection with the 4d Vasiliev theory is discussed.