Background:
let $L \subset \mathbb{Q}^n$ be a lattice (i.e. a finitely generated $\mathbb{Z}$-module). Then $L$ has a reduced basis, that is, a $\mathbb{Z}$-basis $v_1, \dots, v_r$ satisfying $\prod_{i=1}^r |\boldsymbol{v}_i| \ll d(L)$ where $d(L)$ is the discriminant of the lattice and the implicit constants in the upper bound depend only on $n$ and $r$. The LLL algorithm is one way to efficiently compute such a basis.
A natural generalization is to replace $\mathbb{Z}$ with the ring of integers of a number field, $\mathcal{O_K}$, and consider finitely generated $\mathcal{O}_K$ modules $\Lambda \subset K^n$. Because $\mathcal{O_K}$ is a dedekind domain rather than a PID things become a bit more complicated. For instance, $\Lambda$ no longer necessarily has a basis at all. Instead, $\Lambda$ has a "pseudo-basis" of the form $\Lambda = \oplus J_i v_i$ where the $J_i$ are fractional ideals of $\mathcal{O}_K$. This paper by Claus Fieker and Damien Stehlé (Short Bases of Lattices over Number Fields) covers pseudo-bases and gives a generalization of the LLL algorithm to this situation, which returns a pseudo-basis satisfying $\prod |v_i | \ll d(\Lambda)$.
My question:
I specifically want to know if, when $\Lambda$ is a free $\mathcal{O}_K$ module, there exists a small $\mathcal{O}_K$-basis for it (i.e. $\Lambda = \oplus \mathcal{O}_K v_i $ and $\prod |v_i| \ll d(\Lambda)$). This doesn't seem to quite follow from the small pseudo-basis of Fieker and Stehlé. Are any other results known in this direction?