For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by $$\ell(\alpha,\beta)=\liminf_{n\rightarrow\infty}n\|n\alpha\|\|n\beta\|.$$ The Littlewood conjecture asserts that, for all $(\alpha,\beta)\in\mathbb{R}^2$, we have $\ell (\alpha,\beta)=0$.

Can anyone see how to prove the (seemingly much weaker) statement that, for all $(\alpha,\beta)\in\mathbb{R}$, $$\inf_{A\in\mathrm{SL}_2(\mathbb{Z})}\ell\left(A\left( \begin{array}{c} \alpha\\ \beta\\ \end{array} \right)\right)=0\quad ?$$ I wouldn't be surprised if this problem has an easy solution, but I haven't yet been able to find one. For the purposes of orientation note that, in the definition of $\ell$ if we replace $\liminf$ by $\inf$, then the problem becomes close to trivial.