  Many recent works address the question of characterizing induced obstructions
to bounded treewidth. In 2022, Lozin and Razgon completely answered this
question for graph classes defined by finitely many forbidden induced
subgraphs. Their result also implies a characterization of graph classes
defined by finitely many forbidden induced subgraphs that are
$(tw,\omega)$-bounded, that is, treewidth can only be large due to the presence
of a large clique. This condition is known to be satisfied for any graph class
with bounded tree-independence number, a graph parameter introduced
independently by Yolov in 2018 and by Dallard, Milani\v{c}, and \v{S}torgel in
2024. Dallard et al. conjectured that $(tw,\omega)$-boundedness is actually
equivalent to bounded tree-independence number. We address this conjecture in
the context of graph classes defined by finitely many forbidden induced
subgraphs and prove it for the case of graph classes excluding an induced star.
We also prove it for subclasses of the class of line graphs, determine the
exact values of the tree-independence numbers of line graphs of complete graphs
and line graphs of complete bipartite graphs, and characterize the
tree-independence number of $P_4$-free graphs, which implies a linear-time
algorithm for its computation. Applying the algorithmic framework provided in a
previous paper of the series leads to polynomial-time algorithms for the
Maximum Weight Independent Set problem in an infinite family of graph classes.
