On the existence of rook equivalent t-cores

For a positive integer t, a partition is said to be a t-core if each of the hook numbers from its Ferrers–Young diagram is not divisible by t. In 1998, Haglund et al. (J. Combin. Theory Ser. A 84 (1) (1998) 9) proved that if t=2,3, or 4, then two distinct t-cores are rook equivalent if and only if they are conjugates. In contrast to this theorem, they conjectured that if t⩾5, then there exists a constant N(t) such that for every positive integer n⩾N(t), there exist two distinct rook equivalent t-cores of n which are not conjugate. Here this conjecture is proven for t⩾12 with N(t)=4 in all cases.