The incomparability-graph shallow-minor star conjecture

Let $G$ be an incomparability graph, and let $s$ be the number of leaves in a largest induced star in $G$. A $t$-shallow minor is a graph obtained from pairwise vertex-disjoint subgraphs of radius at most $t$ by contracting each subgraph to one vertex. Let $s_t$ denote the number of leaves in a largest induced star in a $t$-shallow minor of $G$, and let ${\hat\beta}_t(G)$ denote the reduced neighborhood clique cover number at depth $t$.

Shallow-minor star conjecture. If $G$ does not have an induced star on $s$ leaves, then, for every $t\ge 0$,

$$s_t=O(t.s).$$

Consequently,

$${\hat\beta}_t(G)=O(t.s).$$

If true, this would give a linear bound on the neighborhood clique cover number for incomparability graphs when the largest induced-star size is fixed. The source does not provide a resolution, so the conjecture remains open.