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We apply a variant of the multigrade efficient congruencing method toestimate Vinogradov's integral of degree $3$ for moments of order $2s$,establishing strongly diagonal behaviour for $1\le s\le 6$. Consequently, themain conjecture is now known to hold for the first time in a case of degreeexceeding $2$.

The cubic case of the main conjecture in Vinogradov's mean value theorem