Polynomial weak colouring-number conjecture for $K^*_{s,t}$-minor-free graphs

For positive integers $s$ and $t$, let $K^*_{s,t}$ be the complete join of $K_s$ and $\overline{K_t}$. A graph is $K^*_{s,t}$-minor-free if it contains no minor isomorphic to $K^*_{s,t}$. For a graph $G$ and integer $r\geqslant1$, let $\operatorname{wcol}_r(G)$ denote its weak $r$-colouring number.

Polynomial weak colouring-number conjecture. There exists a function $f$ such that for every $K^*_{s,t}$-minor-free graph $G$ and every $r\geqslant1$,

$$\operatorname{wcol}_r(G)\leqslant f(s,t)\,r^s.$$

The conjecture would generalise the displayed bounds for $K^*_{2,t}$- and $K^*_{3,t}$-minor-free graphs and would give polynomial weak colouring-number bounds for these minor-closed graph classes. The source gives no resolution status.