The incomparability-graph clique-separator conjecture

Let $p$ be fixed, let $t$ be a nonnegative integer, and let $G$ be an incomparability graph with no $K_{p,p}$ as a $t$-shallow minor. Let $C$ be a clique cover of $G$, and write $|C|$ for its number of cliques. A clique separator is a set of cliques whose removal separates $G$ into two subgraphs.

Clique-separator conjecture. There is a clique cover $C$ in $G$ such that removing

$$O\left(\sqrt{|C|}\right)$$

cliques from $C$ separates $G$ into two subgraphs, each of which can be covered by at most

$$\frac{2|C|}{3}$$

cliques from $C$.

Such a separator would provide a quantitative separation property for incomparability graphs excluding a fixed balanced complete bipartite graph as a shallow minor. The source presents this as a conjecture and gives no resolution, so it remains open.