The comparison conjecture for $A_\alpha$-eigenvalue bounds

Let $G$ be a simple graph, with maximum degree $\Delta$, minimum degree $\delta$, and let $\alpha\in[0,1]$. Comparison conjecture. One has

$$\alpha(\Delta+\delta)+\sqrt{\alpha^2(\Delta-\delta)^2+4\Delta(1-\alpha)^2} \geq\alpha(\Delta+1)+\sqrt{\alpha^2(\Delta+1)^2+4\Delta(1-2\alpha)}.$$

The inequality asserts that the lower bound involving both the maximum and minimum degrees is at least the previously known bound involving only the maximum degree. It is proved for regular graphs and was observed computationally for non-regular graphs with minimum degree greater than one; the general claim remains open.