Quantitative Dickson conjecture for sublattices

Let $L$ be a sublattice of $\mathbb{Z}^t$, let $N$ be large, let $K\subseteq[-N,N]^t$ be convex with volume $\gg N^t$, and let $a\in\mathbb{Z}^t$ have size $O(N)$. Assume that all local obstructions to the Zariski density of prime points in $a+L$ are passed. With $\Lambda$ the von Mangoldt function and $\Lambda^{\otimes t}(n_1,\ldots,n_t)=\prod_{i=1}^t\Lambda(n_i)$, Dickson's conjecture. One expects

$$\frac{1}{|L\cap K|}\sum_{\vec{n}\in L\cap K}\Lambda^{\otimes t}(a+\vec{n})=\prod_p\alpha_p(a+L)+o(1).$$

This is a quantitative form of Dickson's conjecture, expressing the expected mean value of the multidimensional von Mangoldt function in terms of the product of local factors. The source states that the Euler product is convergent; the assertion is not proved in the generality given here.