Conjecture on rational approximations with prescribed numbers of prime divisors

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

$$|\alpha-a/q|\leq 1/(qX).$$

Prescribed-prime-divisors conjecture. There is a rational number $A/Q$ such that $Q\leq X^{1+\epsilon}$, $\omega(Q)=n$, and

$$\left|\alpha-\frac{A}{Q}\right|\ll_\epsilon\frac{1}{qX^{1-\epsilon}}.$$

This asserts that a rational approximation can be replaced by one with a prescribed number of distinct prime divisors in its denominator, with only a small loss in the approximation exponent. The supplied text gives heuristic motivation and related corollaries, but no resolution.