The $C_4$-free analogue of Vizing's conjecture

Let $G$ be a $C_4$-free graph of maximum degree $\Delta$, where $C_4$ denotes a cycle of length four.

The $C_4$-free analogue of Vizing's conjecture. It holds that

$$\chi(G) \le \left\lceil \frac{\Delta+1}{2}\right\rceil+1.$$

This is posed as a related problem in the concluding remarks. The paper gives no resolution, so the analogue remains open.