Almost-sure local-global convergence of growing random regular graphs

Let $(G_n)$ be a growing sequence of random $d$-regular graphs, meaning that the number of vertices tends to infinity. A sequence is local-global convergent when its colored local statistics converge for every finite coloring and every fixed neighborhood radius.

Local-global convergence conjecture. A growing sequence of random $d$-regular graphs is local-global convergent with probability one.

This is stated as a weaker replacement for the disproved Bernoulli-sequence conjecture. The source explicitly says that it remains unsolved.