Characterize graphs with bounded path independence number

Determine whether there exists a function f such that, for all positive integers t and h, every graph G with neither K_{t,t} nor the complete binary tree T_h as an induced minor has path independence number at most f(t,h).

Background

The paper studies path independence number, the minimum integer k for which a graph has a path-decomposition whose bags all have independence number at most k. This parameter is closed under taking induced minors, so any graph containing T_h or K_{t,t} as an induced minor has path independence number that is correspondingly large.

The authors conjecture that these two induced-minor obstructions are the only ones needed to force unbounded path independence number. The conjecture would extend a result of Chudnovsky, Hatzel, Scott, and others from bounded pathwidth to bounded path independence number for graphs excluding K_c as an induced subgraph.