Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence. Is it true that $$\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq 5^{-1/2}\approx 0.447?$$ A conjecture of Newman.

This was proved by Chung and Graham [ChGr84], who in fact prove that $$\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq \frac{1}{c}\approx 0.3944$$ where $$c=1+\sum_{k\geq 1}\frac{1}{F_{2k}}=2.5353705\cdots$$ and $F_m$ is the $m$th Fibonacci number.