Bounded tree-independence when excluding a star (induced) and a planar graph (induced minor)

Establish whether, for every positive integer s and every planar graph H, the class of graphs that exclude the star K_{1,s} as an induced subgraph and exclude H as an induced minor has bounded tree-independence number.

Background

This conjecture generalizes the {P_r, K_{t,t}}-free conjecture by replacing the bipartite graph with an arbitrary planar graph (excluded as an induced minor) and the path with a star (excluded as an induced subgraph).

A weakening giving only a polylogarithmic upper bound on tree-independence number was recently proved, but the full boundedness conjecture remains open.

References

Let us also mention a conjecture of Dallard et al. generalizing \zcref{conj:PrKtt}, which states that excluding a star as an induced subgraph and a planar graph as an induced minor results in a graph class with bounded tree-independence number.

— Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique  (2604.01999 - Chudnovsky et al., 2 Apr 2026) in Section 1 (Introduction)