The bounded tree-independence conjecture for $\{P_k,K_{t,t}\}$-free graphs

For positive integers $k$ and $t$, let a graph be $\{P_k,K_{t,t}\}$-free if it contains no induced subgraph isomorphic to the $k$-vertex path $P_k$ or to $K_{t,t}$.

Bounded tree-independence conjecture. For any two positive integers $k$ and $t$, the class of $\{P_k,K_{t,t}\}$-free graphs has bounded tree-independence number.

The conjecture is open already in this special case, as stated in the source.