Second-largest signless Laplacian spectral radius conjecture for uniform supertrees

Let $S_{2}(m,d,k)$ be the $k$-uniform supertree obtained from a loose path of diameter $d$ by attaching $m-d$ edges at the vertex $v_{\lfloor d/2\rfloor}$. Let $\mathbb{S}(m,d,k)$ denote the family of $k$-uniform supertrees with $m$ edges and diameter $d$, and let $q(G)$ be the signless Laplacian spectral radius of $G$. Second-largest signless Laplacian spectral radius conjecture. If $d\geq 4$ and $m\geq d+1$, then $S_{2}(m,d,k)$ has the second largest signless Laplacian spectral radius in $\mathbb{S}(m,d,k)$. This conjecture proposes the second extremal member of the family for the signless Laplacian spectral radius; the surrounding text indicates that determining this quantity requires new methods and techniques, so its resolution is not supplied here.