On a generalisation of the Dipper-James-Murphy conjecture

Let r,n be positive integers. Let e be 0 or an integer bigger than 1. Let v₁,...,v_r∈Z/eZ and K_r(n) be the set of Kleshchev r-partitions of n with respect to (e;Q), where Q:=(v₁,...,v_r). The Dipper-James-Murphy conjecture asserts that K_r(n) is the same as the set of (Q,e)-restricted bipartitions of n if r=2. In this paper we consider an extension of this conjecture to the case where r>2. We prove that any multi-core Λ=(Λ⁽¹⁾,...,Λ^(r)) in K_r(n) is a (Q,e)-restricted r-partition. As a consequence, we show that in the case e=0, K_r(n) coincides with the set of (Q,e)-restricted r-partitions of n and also coincides with the set of ladder r-partitions of n.