Dickson's conjecture for affine sublattices

Let $L$ be an affine sublattice of $\mathbb{Z}^t$. For each prime $p$, let $\alpha_p(L)$ be the local factor defined from the image $L_p$ of $L$ in $\mathbb{Z}_p^t$; a local obstruction is a prime $p$ for which $\alpha_p(L)=0$. Dickson's conjecture. If all local obstructions to the Zariski density of prime points in $L$ are passed, then $L\cap\mathcal{P}^t$ is Zariski dense in $L$. This generalizes the prime number theorem in arithmetic progressions from affine lines to affine sublattices and is attributed here to Dickson, in the formulation of Bourgain, Gamburd, and Sarnak. The conjecture is open in general, although the arithmetic-progression case follows from the prime number theorem and the case of arithmetic progressions of arbitrary length is a theorem of Green and Tao.