Towards BAD conjecture

For $\alpha, \beta, \delta \in [0,1], \alpha +\beta = 1$ we consider sets $${\rm BAD}^* (\alpha, \beta ;\delta) = \left\{\xi = (\xi_1,\xi_2) \in [0,1]^2: ,\inf_{p\in \mathbb{N}} \max \{(p\log(p+1))^\alpha ||p\xi_1||, (p\log (p+1))^\beta ||p\xi_2||\} \ge \delta \right\}.$$ We prove that for different $(\alpha_1,\beta_1), (\alpha_2,\beta_2), \alpha_1 +\beta_1 = \alpha_2 +\beta_2 = 1$ and $\delta$ small enough $${\rm BAD}^* (\alpha_1, \beta_1 ;\delta) \bigcap {\rm BAD}^* (\alpha_2, \beta_2 ;\delta) \neq \varnothing .$$ Our result is based on A. Khintchine's construction and an original method due to Y. Peres and W. Schlag.