The admissible sums-of-two-squares $k$-tuple conjecture

Let $\mathbb{S}$ be the set of integers expressible as a sum of two squares. For $k\geq1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with $\lvert\boldsymbol{h}\rvert=k$, say that $\boldsymbol{h}$ is $\mathbb{S}$-admissible when, for every prime $p\not\equiv1\pmod4$, there exists $n\in\mathbb{Z}$ such that $n+\boldsymbol{h}\subseteq S_p$, with $S_p$ the relevant local set. The admissible sums-of-two-squares $k$-tuple conjecture. If $\boldsymbol{h}$ is $\mathbb{S}$-admissible, then there exist infinitely many integers $n$ for which $n+\boldsymbol{h}\subseteq\mathbb{S}$. This is a restatement of the qualitative version of the paper's main $k$-tuple conjecture, with local admissibility replacing the equivalent condition that the singular series be nonzero.