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

Let $\mathbb{S}$ be the set of integers expressible as a sum of two squares. For $k\geq 1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with $\lvert\boldsymbol{h}\rvert=k$, call $\boldsymbol{h}$ $\mathbb{S}$-admissible if, for every prime $p\not\equiv1\pmod 4$, there is an integer $n$ such that $n+\boldsymbol{h}\subseteq S_p$, where $S_p$ is the corresponding local set. The qualitative 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 the qualitative, infinitude formulation of the preceding asymptotic conjecture; it remains open in general.