Rationally conditioned two-rational approximation conjecture

Let $\epsilon>0$ be any small positive real number and let $N\geq 1$. Suppose that $\alpha$ has a rational approximation $a/q$ with integers $a,q$, $1\leq q\leq N^2$, $(a,q)=1$, and

$$\left|\alpha-\frac aq\right|\leq\frac{1}{qN^2}.$$

The case $q\leq N$ is immediate by taking $a_1/q_1=a/q$ and $a_2/q_2=0/1$; thus one may restrict attention to $q>N$. Rationally conditioned two-rational approximation conjecture. There exist integers $a_1,a_2$ and $1\leq q_1,q_2\leq N$ such that

$$\left|\alpha-\frac{a_1}{q_1}-\frac{a_2}{q_2}\right|\ll_{\epsilon}\frac{1}{qN^{2-\epsilon}}.$$

This says that a sufficiently good rational approximation should yield an almost equally good approximation by a sum of two rationals; the paper notes that the bound is best possible apart from the factor involving $\epsilon$.