Invertibility conjecture for powers of lattices in rank-$n$ algebras

Let $\mathcal{O}_K$ be a Dedekind domain, let $L$ be an algebra of rank $n$ over $\mathcal{O}_K$, and let $\mathfrak c$ be a lattice in $L$. Let the common endomorphism ring be the endomorphism ring shared by $\mathfrak c^{n-1}$ and all higher powers of $\mathfrak c$.

Lattice-power invertibility conjecture. The lattice $\mathfrak c^{n-1}$ and every higher power $\mathfrak c^m$ with $m\geq n-1$ are invertible in their common endomorphism ring.

The paper proves the rank-three case needed earlier, but does not establish the stated general-rank assertion.