Dallard et al.'s forbidden-path and biclique conjecture for tree-independence number

For a family $\mathcal{F}$ of graphs, a graph $G$ is $\mathcal{F}$-free if no induced subgraph of $G$ is isomorphic to a graph in $\mathcal{F}$. Let $P_r$ be the path with $r$ vertices and let $K_{s,t}$ be the complete bipartite graph with parts of sizes $s$ and $t$, respectively. Dallard et al.'s conjecture. For any two positive integers $r$ and $t$, the class of $\{P_r,K_{t,t}\}$-free graphs has bounded tree-independence number. A weakening giving a polylogarithmic bound is known, and the conjecture has been confirmed for $r\leq 4$, but it remains open in general.