The induced Menger conjecture for $K_{1,d}$-free graphs

Let $k,d$ be positive integers. Let $G$ be a $K_{1,d}$-free graph, and let $A,B\subseteq V(G)$ be disjoint. A set $S$ is a separator between $A$ and $B$ if deleting $S$ separates $A$ from $B$; write $\alpha(S)$ for the independence number of the subgraph induced by $S$. An $A$-$B$ path has one endpoint in $A$ and one in $B$.

Induced Menger conjecture. There exists a function $f:\mathbb{N}^2\rightarrow\mathbb{N}$ such that there exists either $k$ pairwise non-adjacent $A$-$B$ paths, or a separator $S$ between $A$ and $B$ with $\alpha(S)<f(k,d)$.

This is proposed as an approximate induced analogue of Menger's theorem, motivated by the role of disjoint paths in proofs of grid theorems. The parser marks the conjecture as disproved; the paper cites counterexamples to proposed induced Menger variants.