Signless Laplacian extremal conjecture for large $K_{s,t}$-minor free graphs

Let $2\leq s\leq t$ and let $G$ be a $K_{s,t}$-minor free graph of sufficiently large order $n$. Here $q(G)$ denotes the signless Laplacian spectral radius, and $F_{s,t}(n)$ is the extremal graph defined in the paper.

Signless Laplacian extremal conjecture.

$$q(G)\leq q(F_{s,t}(n))$$

with equality if and only if $G=F_{s,t}(n)$.

This conjecture proposes the signless Laplacian analogue of Tait's conjecture for the spectral radius of large $K_{s,t}$-minor free graphs. The claim is proved in the paper for the cases treated there, but its stated general form for all $2\leq s\leq t$ and sufficiently large $n$ remains open.