Bounded-path partition 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 connected partition of width $s-1$ is a partition $H_1,\dots,H_\ell$ into connected subgraphs with quotient width $s-1$ in the sense used by the source. A shortest path is a path shortest in the indicated vertex-deleted graph.

Bounded-path partition conjecture. For all $t\geqslant s\geqslant1$, there exists an integer $p$ such that every $K^*_{s,t}$-minor-free graph $G$ has a connected partition $H_1,\dots,H_\ell$ with width $s-1$, such that for $i\in[\ell]$,

$$V(H_i)=V(P_{i,1})\cup\dots\cup V(P_{i,p_i}),$$

where $p_i\leqslant p$ and each $P_{i,j}$ is a shortest path in

$$G-\Bigl(\bigl(V(H_1)\cup\dots\cup V(H_{i-1})\bigr)\cup\bigl(V(P_{i,1})\cup\dots\cup V(P_{i,j-1})\bigr)\Bigr).$$

The source says that this conjecture would imply the polynomial weak colouring-number conjecture together with the preceding BFS-colouring lemma. Its resolution status is not given.