Polylogarithmic tree-independence conjecture for induced-minor-free graphs

For a positive integer $t$, let $K_{t,t}$ be the complete bipartite graph, and let $W_{t\times t}$ be the $t\times t$-wall. An induced minor is a graph obtained by taking an induced subgraph and contracting pairwise vertex-disjoint connected subgraphs. The tree independence number is the minimum, over all tree decompositions, of the maximum independence number of a bag.

Polylogarithmic tree-independence conjecture. For every positive integer $t$, there is an integer $d=d(t)$ such that for every $n\ge2$, every $n$-vertex graph with no induced minor isomorphic to $K_{t,t}$ or to $W_{t\times t}$ has tree independence number at most $\log^d n$.

This is presented as a further goal toward understanding induced-minor obstructions to polylogarithmic tree independence. The source describes the preceding conjecture and the methods developed there as promising steps, but gives no resolution.