Dickson's generalised Hardy–Littlewood conjecture for affine-linear forms

Let $N,d,t,L$ be positive integers, let $\Psi=(\psi_1,\ldots,\psi_t)$ be a system of affine-linear forms with $\|\Psi\|_N\leqslant L$, and let $K\subset[-N,N]^d$ be a convex body. Define the archimedean factor $\beta_\infty$ and local factors $\beta_p$ as in the paper. Generalised Hardy–Littlewood conjecture. One has

$$\sum_{n\in K\cap\mathbb{Z}^d}\prod_{i\in[t]}\Lambda(\psi_i(n))=\beta_\infty\prod_p\beta_p+o_{t,d,L}(N^d).$$

Here $\Lambda$ is the von Mangoldt function, and the product is over primes. This is the expected asymptotic for prime values of a system of affine-linear forms, incorporating both archimedean and local congruence factors. The conjecture remains open in general, although the paper proves it conditionally for systems of finite complexity from the inverse Gowers-norm and Möbius–nilsequence conjectures.