The DMT conjecture for regular representations of central division algebras

Let ${\mathcal D}$ be an index $n$ ${\mathbb Q}$-central division algebra, let $\Lambda\subset {\mathcal D}$ be an order, and let $\psi_{reg}$ denote its regular representation in $M_n({\mathbb C})$. Regular-representation DMT conjecture. If ${\mathcal D}$ is ramified at the infinite prime, then $\psi_{reg}(\Lambda)$ achieves the DMT upper bound of Theorem

. If ${\mathcal D}$ is not ramified at the infinite \prime, then $\psi_{reg}(\Lambda)$ achieves the DMT of Theorem

. This conjecture asserts that the regular-representation lattice has the same DMT as the conjugate absolute representation, despite not necessarily lying directly in a real or quaternionic matrix subspace; the source says that its proof has eluded the authors.