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

Let $k,d$ be positive integers. The Cartesian product $G\openbox H$ is the graph with vertex set $V(G)\times V(H)$ in which $(u_1,v_1)$ and $(u_2,v_2)$ are adjacent exactly when either $u_1=u_2$ and $v_1v_2\in E(H)$, or $v_1=v_2$ and $u_1u_2\in E(G)$. A $k$-double wheel is $C\openbox P_2$, where $C$ is a cycle on $k$ vertices. A graph is $K_{1,d}$-free if it has no induced $K_{1,d}$, and $\operatorname{\alpha\text{-}\mathsf{tw}}(G)$ denotes its tree-independence number.

Double-wheel conjecture. There exists a function $f:\mathbb{N}^2\rightarrow\mathbb{N}$ such that every $K_{1,d}$-free graph $G$ with $\operatorname{\alpha\text{-}\mathsf{tw}}(G)\geq f(k,d)$ contains the $k$-double wheel as an induced minor.

The double wheel is proposed as an intermediate structure toward a cylindrical induced grid, obtained by closing a ladder into a cycle. The paper notes that proving the conjecture would require new methods, and its status remains open.