Determine the asymptotically sharp chromatic-number base

Determine whether the lower-bound base 2 or the upper-bound base ]3\sqrt{3}/2 is the sharp asymptotic base for the chromatic number of unit-distance graphs in high-dimensional Euclidean spaces.

Background

The paper proves the asymptotic bounds $$(2+o(1))^d\le \chi(\mathbb{R}^d)\le (3\sqrt{3}/2+o(1))^d$$. The authors explicitly conjecture that one of these two exponential bases is sharp, but do not specify which one. Resolving this would determine the leading exponential growth rate of the chromatic number of high-dimensional Euclidean unit-distance graphs.