Existence of a finite dimension attaining the limiting 1-independent percolation threshold

For $d\geq 3$, let $p_{1,c}(\mathbb{Z}^d)$ denote the critical probability for $1$-independent percolation on the integer lattice. High-dimensional percolation conjecture. There exists $d\geq 3$ such that

$$p_{1,c}(\mathbb{Z}^d)=4-2\sqrt{3}.$$

This is presented as a stronger conjecture following the hypercube conjecture and the known lower bound $p_{1,c}(\mathbb{Z}^d)\geq 4-2\sqrt{3}$. Its resolution would settle the limiting-value question of Balister and Bollobás.