Borel chromatic-number conjecture for graphs of subexponential growth

Let $G$ be a Borel graph of subexponential growth. Write $a(G)$ for its asymptotic separation index and $\chi_{\mathrm{B}}(G)$ for its Borel chromatic number.

Borel chromatic-number conjecture.

$$\chi_{\mathrm{B}}(G) \leqslant \max\{2a(G),3\}.$$

This conjecture would sharpen the general bound $\chi_{\mathrm{B}}(G) \leqslant 2a(G)+1$ and, when $a(G)\geqslant 2$, give the proposed bound $\chi_{\mathrm{B}}(G)\leqslant 2a(G)$. Its status is not resolved in the supplied material.