Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique

We show that for every positive integer $t\ge 2$ there exists an integer $s$ such that every graph that contains no induced subgraph isomorphic to either the $6$-vertex path or the $(2,t)$-biclique, the complete bipartite graph $K_{2,t}$, has tree-independence number at most $s$. This result makes partial progress on a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht.