The forbidden biclique and path conjecture for tree-independence number

For positive integers $d$ and $s$, a graph is $\{K_{d,d},P_s\}$-free if it has no induced subgraph isomorphic to $K_{d,d}$ or the $s$-vertex path $P_s$. The biclique-and-path conjecture. For any two positive integers $d$ and $s$, the class of $\{K_{d,d},P_s\}$-free graphs has bounded tree-independence number. This is a special case of the finite-forbidden-subgraph conjecture. The case with $P_4$ is discussed as known, while the full statement for arbitrary $d$ and $s$ remains open.