Littlewood's conjecture on simultaneous rational approximation to a pair of real numbers, $$ \liminf_{n \in \mathbb{N}} \, n \cdot \mathrm{dist}(n\alpha,\mathbb{Z}) \cdot \mathrm{dist}(n\beta, \mathbb{Z}) = 0, $$ has remained one of the outstanding open problems in Diophantine Approximations. A natural extension of it, considered by Cassels and Swinnerton-Dyer in the 1950s, turns out exactly equivalent to a conjectured rigidity property in higher rank hyperbolic homogeneous dynamics: all bounded orbits of the Weyl chamber flow on $\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$ are closed (or, one could say, 'periodic'). The best result on the problem, the Einsiedler-Katok-Lindenstrauss theorem that the conjecture holds for all pairs $(\alpha,\beta)$ outside of a set of zero Hausdorff dimension, has come from this ergodic theory avenue. However, special cases such as $(\sqrt{2},\sqrt{3})$ have remained unsolved.
It is also possible to extend the problem in a direction completely disjoint from homogeneous dynamics. Observe that if $(\alpha,\beta)$ is a counterexample to Littlewood's conjecture then, for a large enough constant $C < \infty$, $$ P_n : \, (x_n,y_n) = (C\{n\alpha\}, C \{n\beta\}) \in \mathbb{R}^2 $$ gives an infinite bounded sequence of points in the plane (or a torus, if one prefers), such that $$ |x_n-x_m| |y_n-y_m| \geq \frac{1}{|n-m|} $$ for all $n \neq m$.
Question: Does such an infinite bounded sequence exist? Or is this extension of Littlewood's conjecture considered anywhere?
Note that, as explained by Ilya Bogdanov solving this question of mine, such a sequence does exist if one weakens the condition to $$ \max(|x_n-x_m|^2, |y_n-y_m|^2) \geq \frac{1}{|n-m|}. $$
