Badziahin–Velani conjecture for the logarithmic spectrum of multiplicatively badly approximable matrices

For $lambda eu0007\mathbb{R}$, let $\operatorname{Mad}^{\lambda}(2,1)$ denote the corresponding logarithmic approximation set in the multiplicative setting. Badziahin–Velani conjecture.

$$\operatorname{Mad}^{\lambda}(2,1)= \begin{cases} \emptyset & \text{if }\lambda<1,\\ \text{full Hausdorff dimension but zero Lebesgue measure set} & \text{if }1\leq\lambda\leq2,\\ \text{full Lebesgue measure set} & \text{if }\lambda>2. \end{cases}$$

This conjecture predicts the analogue of the additive logarithmic zero–one transition after removing one logarithmic factor. The source presents it as a conjecture of Badziahin and Velani; no resolution is supplied here.