Conjecture on nonvanishing determinant conditions for lattice codes

Let $n\geq m$, let $\mathcal{L}$ be a $2mn$-dimensional lattice in $M_n(\mathbb{C})$, and suppose that

$$\Delta_m(\mathcal{L})\neq 0.$$

Here $\Delta_m(\mathcal{L})$ denotes the infimum of the products of the smallest $m$ singular values of nonzero elements of $\mathcal{L}$. Nonvanishing determinant condition conjecture. The conditions of Corollary can be satisfied only when either $m=n$ or when $n=2$ and $m=1$. This concerns when a $2mn$-dimensional lattice code in $M_n(\mathbb{C})$ can satisfy the nonvanishing product-of-singular-values condition required for approximate universality with $m$ receive antennas. The claim is presented as conjectural, and no resolution is given in the source.