Generalized Montgomery conjecture for dynamic chromatic number

Let $G$ be a nontrivial connected graph. Write $\Delta(G)$ and $\delta(G)$ for its maximum and minimum degrees, respectively, and let $\chi(G)$ and $\chi_2(G)$ denote its chromatic and dynamic chromatic numbers. Generalized Montgomery conjecture.

$$\chi_2(G)-\chi(G)\leq 2\left\lceil\frac{\Delta(G)}{\delta(G)}\right\rceil.$$

This conjecture generalizes Montgomery's regular-graph claim, since for a regular graph the degree ratio is $1$. The supplied text gives no resolution status.