Conjecture on approximation by rationals with prime denominators

Let $\xi$ be a real number, let $a$ be an integer, and let $p$ be a prime number. Consider rational approximations $a/p$ whose error is measured by

$$\left|\xi-\frac{a}{p}\right|.$$

Prime-denominator approximation conjecture. For every real $\xi$, there exists a constant $c(\xi)$ such that the inequality

$$\left|\xi-\frac{a}{p}\right|<\frac{c(\xi)\ln p}{p^2}$$

with $p$ prime has infinitely many solutions in integers $a$ and prime numbers $p$. The claim is presented as a conjectural consequence of the density of primes and a constructive procedure; the supplied text gives no proof or resolution.