  A decade ago two groups of authors, Karasev, Hubard and Aronov, and
Blagojevi\'c and Ziegler, have shown that the regular convex partitions of a
Euclidean space into $n$ parts yield a solution to the generalised Nandakumar
and Ramana-Rao conjecture when $n$ is a prime power. This was obtained by
parametrising the space of regular equipartitions of a given convex body with
the classical configuration space.
  Now, we repeat the process of regular convex equipartitions many times, first
partitioning the Euclidean space into $n_1$ parts, then each part into $n_2$
parts, and so on. In this way we obtain iterated convex equipartions of a given
convex body into $n=n_1...n_k$ parts. Such iterated partitions are parametrised
by the (wreath) product of classical configuration spaces. We develop a new
configuration space -- test map scheme for solving the generalised Nandakumar
\& Ramana-Rao conjecture using the Hausdorff metric on the space of iterated
convex equipartions.
  The new scheme yields a solution to the conjecture if and only if all the
$n_i$'s are powers of the same prime. In particular, for the failure of the
scheme outside prime power case we give three different proofs.
