Halting-realizability conjecture for cardinal-inequality \texorpdfstring{$\otimes$}{⊗}-graphs

Let $\mathcal{A}$ be an algorithm and $x$ an input. A $\otimes$-graph with cardinal inequalities is an $\otimes$-graph $\mathcal{G}$ equipped with cardinal inequalities between its places, and it is realizable when such a graph has a realization satisfying those inequalities. Halting-realizability conjecture. For any algorithm $\mathcal{A}$ and input $x$, one can construct a $\otimes$-graph $\mathcal{G}$ with cardinal inequalities such that

$$\mathcal{A}\text{ terminates on input }x\quad\Longleftrightarrow\quad\mathcal{G}\text{ is realizable}.$$

This would give a direct reduction of the halting problem to realizability for cardinal-constrained $\otimes$-graphs, complementing the paper's observation that realizability for such graphs is undecidable because the Hilbert's tenth problem is reducible to it.