Extremal spectral-radius conjecture for graphs excluding a complete bipartite minor

Let $2\leq s\leq t$, let $0\leq p<t$, and write

$$n=s-1+kt+p.$$

Here, $K_r$ denotes the complete graph on $r$ vertices, $kK_t$ denotes the disjoint union of $k$ copies of $K_t$, and $G\vee H$ denotes the join of two graphs. Extremal spectral-radius conjecture. For $n$ sufficiently large, the $n$-vertex graph of maximum spectral radius among graphs that do not contain $K_{s,t}$ as a minor is

$$K_{s-1}\vee (kK_t\mathbin{+}K_p).$$

This conjectures the extremal construction in the cases where $t$ does not divide $n-s+1$, extending the preceding equality characterization when the remainder is zero. The conjecture concerns the structure of graphs excluding a complete bipartite minor and their maximum spectral radius; the supplied text gives no resolution.