The exponential convergence conjecture for triangle-free unit-distance digraphs

For $d\in\mathbb{N}$, let $G(d)$ be the infimum of $\bar{t}(G)/|E(G)|$ over all finite unit-distance digraphs $G$ in $\mathbb{R}^d$ with $|E(G)|>0$, where $\bar{t}(G)$ is the maximum number of edges in a subgraph without any directed cyclic triangles. A digraph is unit-distance if every edge $xy$ satisfies $\lVert x-y\rVert_2=1$.

Exponential convergence conjecture. There exist absolute constants $c>0$ and $d_0\in\mathbb{N}$ such that, for all $d\geqslant d_0$,

$$G(d)\leqslant \frac{1}{2}+\exp(-cd).$$

This conjecture would give an exponentially sharp upper bound for the extremal density of $\Delta$-free subsets of high-dimensional spheres, complementing the paper's polynomial upper bound and its connection between such sets and triangle-free unit-distance digraphs.