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A new concept of moderate non-locality in higher-spin gauge theory isintroduced. Based on the recently proposed differential homotopy approach, amoderately non-local scheme, that issofter than those resulting from theshifted homotopy approach available in the literature so far, is worked out inthe mixed $\eta\bar\eta$ sector of the Vasiliev higher-spin theory. Tocalculate moderately non-local vertices $\Upsilon^{\eta\bar \eta}(\omega,C,C,C)$ for all ordering of the fields $\omega$ and $C$ we apply aninterpolating homotopy, that respects the moderate non-locality in theperturbative analysis of the higher-spin equations.

Moderately non-local $\eta \bar\eta$ vertices in the $AdS_4$ higher-spin gauge theory