Tree-independence number of $P_5$-free graphs with no large bicliques

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\mathrm{\ell },\mathrm{\ell }}$ forces tree-independence number at least $\mathrm{\ell }$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\mathrm{\ell }$, every $\{P_t,K_{\mathrm{\ell },\mathrm{\ell }}\}$-free graph has bounded tree-independence number. We prove this conjecture for $t=5$ by showing that every $\{P_5,K_{\mathrm{\ell },\mathrm{\ell }}\}$-free graph has tree-independence number at most $4\mathrm{\ell }$. We also obtain related bounds for the weaker parameter of $\alpha$-degeneracy.