The k-creature minimal separator conjecture

For an integer $k \geq 3$, a graph $G$ is a $k$-creature if

$$V(G)=A\cup B\cup\\{x_1,\dots,x_k\\}\cup\\{y_1,\dots,y_k\\}$$

such that $G[A]$ and $G[B]$ are connected, $A$ is anticomplete to $B$, for $i=1,\dots,k$, $x_i y_i\in E(G)$, $x_i$ has a neighbor in $A$ and is anticomplete to $B$, $y_i$ has a neighbor in $B$ and is anticomplete to $A$, and for $1\leq i,j\leq k$ with $i\ne j$, $x_i y_j\notin E(G)$. A minimal separator is a vertex set that minimally separates some pair of vertices. $k$-creature conjecture. There exists $f:\mathbb{N}\to\mathbb{N}$ such that if no induced subgraph of $G$ is a $k$-creature, then $G$ has at most $|V(G)|^{f(k)}$ minimal separators. This is proposed as a stronger version of the paper's main theorem: excluding induced $k$-creatures is conjectured to imply a polynomial bound on minimal separators, but the source provides no resolution.