Badly approximable vectors in affine subspaces: Jarník-type result

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