Cavers et al.'s maximum skew-spectral radius conjecture for odd-cycle graphs

Let $G$ be a simple graph with no even cycle, called an odd-cycle graph, and let $\rho_s(G)$ denote its maximum skew-spectral radius. For each positive integer $n$, let $H_n$ be the odd-cycle graph of order $n$ with one vertex of degree $n-1$ and size

$$m=\left\lfloor\frac{3(n-1)}{2}\right\rfloor.$$

Cavers et al.'s conjecture. If $G$ is an odd-cycle graph of order $n$, then

$$\rho_s(G)\leq\rho_s(H_n),$$

and equality holds if and only if $G\cong H_n$. The equivalent formulation replaces $\rho_s(G)$ by the maximum matching root $t(G)$. The conjecture asks for the unique extremal graph, up to isomorphism, among odd-cycle graphs of fixed order.