Minimum spectral radius conjecture for graphs with odd order and prescribed domination number

Let $n\geq3$ be odd, and let $\mathbb{G}_{n,\lfloor\frac{n}{2}\rfloor}$ denote the class of graphs on $n$ vertices with domination number $\lfloor\frac{n}{2}\rfloor$. Let $H'$ be the tree obtained from $P_{\lfloor\frac{n}{2}\rfloor}\circ K_1$ by subdividing once a pendant edge on its diameter, and write $H=H'$ as in the conjecture. Here $\rho(G)$ denotes the spectral radius of a graph $G$.

Minimum spectral radius conjecture. For every graph $G\in\mathbb{G}_{n,\lfloor\frac{n}{2}\rfloor}$,

$$\rho(G)\geq\rho(H),$$

and equality holds if and only if $G\cong H'$.

The preceding even-order result identifies the minimizer as $P_{n/2}\circ K_1$. The odd-order analogue is presented as not easy to prove, and no resolution is supplied in the given text.