  We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with
$a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$.
These are the most extremal examples currently known in the $abc$ conjecture,
thereby providing a new lower bound on the tightest possible form of the
conjecture. This builds on work of van Frankenhuysen (1999) who proved the
existence of examples satisfying the above bound with the constant $6.068$ in
place of $6.563$. We show that the constant $6.563$ may be replaced by
$4\sqrt{2\delta/e}$ where $\delta$ is a constant such that all full-rank
unimodular lattices of sufficiently large dimension $n$ contain a nonzero
vector with $\ell_1$ norm at most $n/\delta$.
