Maximal Hausdorff dimension conjecture for badly approximable complex matrices

Let $m,n$ be positive integers, let $\textbf{Bad}_{\mathbb{C}}(m,n)\subset\mathbb{C}^{mn}$ denote the set of badly approximable points, and let $K\subset\mathbb{C}^{mn}$ be any compact subset. The Hausdorff dimension is denoted by $\dim$. Maximal-dimension conjecture.

$$\dim\bigl(\textbf{Bad}_{\mathbb{C}}(m,n)\cap K\bigr)=\dim K.$$

This conjectures that badly approximable points have full Hausdorff dimension inside every compact subset of the ambient complex matrix space. The cited framework establishes analogous maximal-dimension results for certain weighted simultaneous-approximation settings, but the source states that it does not apply to the dual setup considered here.