Degree 3 dilation conjecture for the infinite square lattice

Let $\Lambda$ be the infinite square lattice, and let $\delta_0(\Lambda,3)$ denote its minimum stretch factor among degree-$3$ spanners. Degree 3 dilation conjecture.

$$\delta_0(\Lambda,3) =(3+2\sqrt2) \, 5^{-1/2} = 2.6065\ldots$$

The paper has established the matching upper bound and conjectures that it is optimal; whether the upper bound can be improved is therefore left open.