Interpolated rational-sum approximation conjecture

Let $\alpha$ be a real number, let $N\geq 1$ be real, and let $0\leq\beta\leq 1$. The integers $a_1,a_2$ and positive integers $q_1,q_2$ satisfy $1\leq q_1,q_2\leq N$. Interpolated rational-sum approximation conjecture. There exist such integers $a_1,a_2,q_1,q_2$ for which

$$\left|\alpha-\frac{a_1}{q_1}-\frac{a_2}{q_2}\right|\leq\frac{1}{(q_1q_2)^\beta N^{2-\beta}}.$$

The claim is known for the endpoint cases $\beta=0$ and $\beta=1$, while the cases $0<\beta<1$ are left as an open problem.