  We enhance the efficient congruencing method for estimating Vinogradov's
integral for moments of order $2s$, with $1\le s\le k^2-1$. In this way, we
prove the main conjecture for such even moments when $1\le s\le
\tfrac{1}{4}(k+1)^2$, showing that the moments exhibit strongly diagonal
behaviour in this range. There are improvements also for larger values of $s$,
these finding application to the asymptotic formula in Waring's problem.
