Heath-Brown's conjecture on primes in arithmetic progressions

Let $A\geq 2$, and let $b$ and $l$ be coprime integers. Heath-Brown's conjecture. There exists a prime $p\equiv b\pmod l$ such that

$$p\ll l\left(\log l\right)^A.$$

This conjecture is used as a conditional input for constructing many $(a,a)$-Carmichael numbers with prescribed greatest common divisor of the numbers $p-1$. The source notes that the exponent $2$ is expected to be essentially optimal, and uses a weakened form to obtain the paper's main conditional result.