Winning conjecture for badly approximable vectors on rational quadratic varieties

Let $X_m(\mathbb{Z})$ denote the integral points on the rational quadratic variety of parameter $m$, and let $BA_{\partial X}^{\psi}(X_m(\mathbb{Z}))$ be the set of points on $\partial X$ badly approximable with respect to $X_m(\mathbb{Z})$ and approximation function $\psi$. Set

$$\psi(t)=t^{-1}.$$

Winning conjecture. For every $m\in\mathbb{Q}$, the set $BA_{\partial X}^{\psi}(X_m(\mathbb{Z}))$ is a (strong) winning subset of $\partial X$.

This conjecture predicts that the badly approximable set is large in the sense of Schmidt games, despite being a null set when $m\ne 0$. Theorem and an analogous example from the literature provide supporting evidence, but the conjecture is not resolved in the supplied text.