  We prove general type results for orthogonal modular varieties associated
with the moduli of compact hyperk\"ahler manifolds of deformation generalised
Kummer type ('deformation generalised Kummer varieties'). In particular, we
consider moduli spaces of deformation generalised Kummer fourfolds with
split-polarisation of degree $2d$. Our main result is that when $d$ is prime or
$2d$ is square-free then the associated modular varieties are of general type
when $d$ exceeds bounds we determine, subject to the existence of certain
low-weight cusp forms for $\operatorname{O}(2,n)$. As a corollary, we conclude
that the corresponding moduli spaces are also of general type.
