Badly approximable vectors in affine subspaces: Jarn\'{\i}k-type result

Consider irrational affine subspace $A\subset \mathbb{R}^d$ of dimension $a$. We prove that the set $$\{\xi =(\xi_1,...,\xi_d) \in {A}:\,\,\, \ q^{1/a}\cdot \max_{1\le i \le d} ||q\xi_i|| \to \infty,\,\,\,\, q\to \infty\}$$ is an $\alpha$-winning set for every $\alpha \in (0,1/2]$