Prime arithmetic progression counting conjecture

Let $k$ be the length of the progression, let $d$ be its common difference, and let $A_{k,d}$ be the adjustment factor defined by the local prime-divisibility probabilities. Prime arithmetic progression conjecture. The number of arithmetic progressions of primes of length $k$ and common difference $d$, beginning with a prime $p\leq N$, is

$$A_{k,d}\int_2^N\frac{dx}{(\log x)^k}\sim\frac{A_{k,d}N}{(\log N)^k}.$$

This specializes the polynomial prime-value heuristic to arithmetic progressions; it remains open in general.