Erdős–Odlyzko–Sárközy conjecture on products of two small primes

Let $k$ be a positive integer and let $a$ be an integer coprime to $k$. The primes $p_1$ and $p_2$ are required to satisfy $p_1,p_2\leq k$. Erdős–Odlyzko–Sárközy conjecture. For all sufficiently large $k$ and $a$ with $(a,k)=1$, there exist primes $p_1,p_2\leq k$ such that

$$p_1p_2\equiv a\pmod{k}.$$

This conjecture is stated by Erdős, Odlyzko and Sárközy and is unreachable with current methods because of the parity problem; various relaxations are known.