Property (T) group factors whose Jones index set equals all positive integers

Using a m\'elange of techniques at the rich intersection of deformation/rigidity theory, finite index subfactor theory, and geometric group theory, we prove the existence of a continuum of property (T) factors that are pairwise non-stably isomorphic and whose Jones index sets consist of all positive integers. These factors are realized as group von Neumann algebras $\mathcal{L}(G)$ associated with property (T) generalized wreath-like product groups $G \in \mathscr{WR}(A, B \curvearrowright I)$ introduced in [CIOS23b], where $A$ is abelian, $B$ is a non-parabolic subgroup of a relatively hyperbolic group with residually finite peripheral structure, and $B \curvearrowright I$ is a faithful action with infinite orbits. Integer index subfactors of $\mathcal{L}(G)$ are constructed from extensions of $G$. This result advances an open question of P. de la Harpe [dlH95].