Asymptotic conjecture for non-degenerate systems of linear equations in primes

Let $s,t$ be positive integers with $t\geqslant s+2$. An $s\times t$ integer matrix $\mathbf{A}$ is non-degenerate if it has rank $s$ and every non-zero vector in its row span over $\mathbb{Q}$ has at least three non-zero entries. The asymptotic conjecture for non-degenerate systems. For fixed $s\geqslant1$, if $t\geqslant s+2$ and $\mathbf{A}$ is non-degenerate, then the Hardy–Littlewood conjecture holds for $\mathbf{A}\mathbf{p}=0$. This is presented as the authors' eventual goal and is generally unresolved, although it includes cases already known for particular systems.