Spectral-radius chromatic bound for triangle-free graphs

Let $G$ be a triangle-free graph, let $\chi(G)$ denote its chromatic number, and let $\rho(G)$ denote its spectral radius. Spectral-radius chromatic bound. Every triangle-free graph satisfies

$$\chi(G)\le(1+o(1))\frac{\rho(G)}{\ln \rho(G)}.$$

The statement is presented as a possible strengthening of the paper's theorem, moving toward bounds conjectured by Harris; the supplied text gives no resolution of it.