  We prove that near-threshold negative energy solutions to the 2D cubic
($L^2$-critical) focusing Zakharov-Kuznetsov (ZK) equation blow-up in finite or
infinite time. The proof consists of several steps. First, we show that if the
blow-up conclusion is false, there are negative energy solutions arbitrarily
close to the threshold that are globally bounded in $H^1$ and are spatially
localized, uniformly in time. In the second step, we show that such solutions
must in fact be exact remodulations of the ground state, and hence, have zero
energy, which is a contradiction. This second step, a nonlinear Liouville
theorem, is proved by contradiction, with a limiting argument producing a
nontrivial solution to a (linear) linearized ZK equation obeying
uniform-in-time spatial localization. Such nontrivial linear solutions are
excluded by a local-viral space-time estimate. The general framework of the
argument is modeled on Merle [29] and Martel & Merle [24], who treated the 1D
problem of the $L^2$-critical gKdV equation. Several new features are
introduced here to handle the 2D ZK case.
