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, Ono, and Sze proved that if $t=2,3$, or $4$, then two distinct $t$-core partitions are rook equivalent if and only if they are conjugates. In contrast to this theorem, they conjectured that if $t\geq 5$, then there exists a constant $N(t)$ such that for every positive integer $n\geq N(t)$, there exist two distinct rook equivalent $t$-core partitions of $n$ which are not conjugate. I prove this conjecture for $t\geq 12$ with $N(t)=4$ in all cases.