Let $K$ be a quintic extension of ${\mathbb Q}$ with ring of integers $R$ and let $p$ be an unramified prime in $K$. Is it true that the number of orders in $R$ of index equal to $p^r$, for some natural number $r$, is less than or equal to the number of subrings with identity of ${\mathbb Z}^5$ of index equal to $p^r$?

I've been staring at Jos Brakenhoff's thesis for a while, but I haven't gotten anywhere. Any advice will be greatly appreciated. Thanks.