Li and Zhang's signless Laplacian conjecture for cacti with given matching number

Let $G$ be a cactus on $n$ vertices with matching number $m$. Let $q_1(G)$ denote the signless Laplacian spectral radius of $G$, and let $H_m^0$ and $H_{m-1}^{n-2m+1}$ be the cacti used in the cited extremal constructions.

Li and Zhang's conjecture. If $n=2m+1$, then

$$q_1(G)\leq\frac{(n+2)+\sqrt{n^2-4n+12}}{2},$$

with equality if and only if $G\cong H_m^0$. If $n\geq2m+2$, then

$$q_1(G)\leq q_1(H_{m-1}^{n-2m+1}),$$

with equality if and only if $G\cong H_{m-1}^{n-2m+1}$, where $q_1(H_{m-1}^{n-2m+1})$ is the largest root of

$$x^3-(n+3)x^2+3nx-4m+4=0.$$

The conjecture proposes the extremal cacti and their exact signless Laplacian spectral radii for the two ranges beyond the perfect-matching case. The supplied paper explains that the original formulas in Li and Zhang's conjecture require revision; this corrected formulation remains a conjecture here.