Liu's modified degree-sequence conjecture for extremal nonregular graphs

Let $\mathcal{G}(n,\Delta)$ be the set of connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attain the maximum spectral radius. Suppose $\Delta\geq 3$ and $G\in\mathcal{G}(n,\Delta)$. For each fixed $\Delta$ and sufficiently large $n$, write the degree sequence of $G$ as $(\Delta,\ldots,\Delta,\delta)$. Liu's modified conjecture. The exceptional degree is

$$\delta=\begin{cases} \Delta-1,&\text{if }\Delta\text{ and }n\text{ are both odd},\\ 1,&\text{if }\Delta\text{ is odd and }n\text{ is even},\\ \Delta-2,&\text{if }\Delta\text{ is even}. \end{cases}$$

This is a modification of the Liu–Li conjecture after the cases $\Delta=3,4$ were confirmed. The source presents the modified statement as a conjecture; its general resolution is not supplied.