Jakobson's variation of the prime $k$-tuples conjecture for sums of two squares

Let $\mathcal{H}^*=\{h_1,h_2,\ldots\}$ be an ordered set of integers. It is admissible if every finite truncation $\{h_1,\ldots,h_k\}$ is admissible, meaning that its reduction modulo every prime $p$ omits at least one residue class. Jakobson's variation of the prime $k$-tuples conjecture. If $\mathcal{H}^*$ is admissible, then there exists an increasing sequence of integers $n_k$ such that, for every $k\geq 1$, the translates $n_k+h_1,\ldots,n_k+h_k$ are sums of two squares. The paper presents this as a variation of the prime $k$-tuples conjecture and states that it has implications for quantum limits on flat tori; the supplied text does not establish it as solved or refuted.