  The ratio
$\mathcal{B}(K_L\to\pi^0\nu\bar\nu)/\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$ of the
branching fractions of kaon decays $K_L\to\pi^0\nu\bar\nu$ and
$K^+\to\pi^+\nu\bar\nu$ has a maximum of about 4.3 under the assumption that
the underlying interactions change isospin by $\Delta I=1/2$. This is referred
to as the Grossman-Nir (GN) bound, which is respected by the standard model
(SM) and by many scenarios beyond it. Recent preliminary results of the KOTO
and NA62 Collaborations searching for these kaon modes seem to imply a
violation of this bound. The KOTO findings also suggest that
$\mathcal{B}(K_L\to\pi^0\nu\bar\nu)$ could be much larger, by nearly two orders
of magnitude, than that predicted in the SM. In this work we study the
possibility of violating the GN bound in an effective field theory approach
with only SM fields. We show that the bound holds, in addition to the original
GN scenarios, whether or not the kaon decays conserve lepton number. We
demonstrate that the inclusion of $\Delta I=3/2$ operators can lead to a
violation of the GN bound and illustrate with an example of how the KOTO
numbers may be reached with a new physics scale of order tens of GeV.
