Subpolynomial treewidth conjecture for induced-minor-free graphs

Let $W_{t\times t}$ be the $t$-by-$t$ hexagonal grid and let $K_{t,t}$ be the complete bipartite graph with both sides of the bipartition of size $t$. For a positive integer $t$, let $\mathcal{C}_t$ be the class of graphs with no induced minor isomorphic to $K_{t,t}$ or $W_{t\times t}$, and let $\mathcal{C}_t^*$ be the subclass consisting of graphs with no clique of size $t$. Here $tw(G)$ denotes the treewidth of a graph $G$. Subpolynomial treewidth conjecture. For every $t\in{\mathbb N}$, there is an integer $d=d(t)$ such that every $n$-vertex graph $G\in\mathcal{C}_t^*$ satisfies

$$tw(G)\leq\log^d n.$$

The conjecture predicts a polylogarithmic treewidth bound for graphs excluding fixed induced minors and a fixed-size clique, which would support efficient or quasi-polynomial algorithms for problems such as Maximum Weight Independent Set on these classes. Its resolution is not supplied in the source context.