  Iwasawa-theoretic descent calculations are particularly delicate in the
presence of additional totally split primes, a situation commonly referred to
as the case of trivial zeroes. We generalise a conjecture of Mazur-Rubin and
Sano that contains the essential information needed in such a scenario to
arbitrary higher-rank Euler systems for $\mathbb{G}_m$, and provide
unconditional evidence for its validity. This includes a proof of the
conjecture for the Euler system of elliptic units, which allows us to derive
new cases of the $p$-part of the equivariant Tamagawa number conjecture for
abelian extensions of an imaginary quadratic field.
