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

Let $\eta,d$ be positive integers. Let $G$ be a $K_{1,d}$-free graph, and let $\operatorname{\alpha\text{-}\mathsf{tw}}(G)$ denote its tree-independence number. A set $S$ is a separator between two vertex sets if deleting $S$ separates them; $\alpha(S)$ denotes the independence number of the subgraph induced by $S$.

Two induced cycles conjecture. There exists a function $f:\mathbb{N}^2\rightarrow\mathbb{N}$ such that every $K_{1,d}$-free graph $G$ with $\operatorname{\alpha\text{-}\mathsf{tw}}(G)\geq f(\eta,d)$ contains two induced cycles $C_1,C_2$ such that $C_1$ and $C_2$ are non-adjacent, and every separator $S$ between $C_1$ and $C_2$ satisfies $\alpha(S)\geq\eta$.

This conjecture would provide an analogue of the paper's path-and-cycle connectivity result with two induced cycles, potentially supporting the double-wheel program. The paper presents it as a possible next step and gives no resolution.