Conjecture on sums of rational approximations with prime denominators

Let $\alpha$ be a real number, let $N\geq 1$, let $\epsilon>0$, and let $n\leq \epsilon\log N$ be a positive integer. Suppose that there are integers $a,q$ with $1\leq q\leq N^n$ and $(a,q)=1$ such that

$$|\alpha-a/q|\leq 1/(qN^n).$$

Sums of rational approximations conjecture. There are integers $a_1,\ldots,a_n$ and distinct primes $q_1,\ldots,q_n$ with $1\leq q_i\leq N$ such that

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

This conjecture predicts an essentially optimal preservation of the quality of a rational approximation while expressing the approximant as a sum with distinct prime denominators. The paper presents related partial results with a weaker exponent, but the supplied text does not establish the conjecture.