The nonseparable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors

In 1967, Kadison asked ``does every type $\mathrm{II}_1$ factor have an orthonormal (with respect to the trace) basis consisting of unitaries?''In a previous paper \cite{HTZ26}, He, Tang, and Zhang resolved Kadison's problem in the separable case. We prove the complementary nonseparable case and thereby resolve Kadison's problem in full. In fact, the basis may be chosen to consist of self-adjoint unitaries. The proof combines a relative norming lemma under small-density constraints, a finite-layer certification scheme ensuring that the relevant Hilbert-space projections are represented by bounded elements of the ambient factor, and a transfinite extension along the density character of $L^2(M,τ)$.