Tait's extremal $A_0$-spectral radius conjecture for $K_{a,b}$-minor-free graphs

Let $a$ and $b$ be integers with $b\geqslant a\geqslant 2$, and write

$$n-1=kb+t,\qquad 0\leqslant t\leqslant b-1.$$

For graphs, let $A_0(G)=A(G)$, let $K_{a-1}\vee (kK_b\cup K_t)$ denote the join of $K_{a-1}$ with the disjoint union of $k$ copies of $K_b$ and $K_t$, and let the $A_0$-spectral radius be the largest absolute value of an eigenvalue of $A_0(G)$. Tait's conjecture. For sufficiently large $n$, the graph $K_{a-1}\vee (kK_b\cup K_t)$ attains the maximum $A_0$-spectral radius among all $K_{a,b}$-minor-free graphs with $n$ vertices.

This is a spectral Turán-type conjecture for excluding a complete bipartite graph as a minor. The supplied text attributes the conjecture to Tait and does not state whether it has been resolved.