Resolvent construction conjecture for asymptotically stationary abstract Klein-Gordon operators

Let $K$ be an abstract Klein-Gordon operator on $L^2(\mathbb{R},\mathcal K)$, and let $R_z(t,s)$ and $\mathcal Z_{\pm,z}^{(\pm)}(t)$ be the evolution and asymptotic subspaces defined in the source. For $\operatorname{Re}(z)\geq0$, suppose that the pair $(\mathcal Z_{+,z}^{(+)}(t),\mathcal Z_{-,z}^{(-)}(t))$ is complementary, and for $\operatorname{Re}(z)\leq0$ equivalently suppose that $(\mathcal Z_{+,z}^{(-)}(t),\mathcal Z_{-,z}^{(+)}(t))$ is complementary. Let the corresponding projections define $E_z(t,s)$ and $G_z(t,s)=-\mathrm{i}E_{z,12}(t,s)$. Resolvent construction conjecture. The operator $G_z$ is bounded on $L^2(\mathbb{R},\mathcal K)$, has dense range and trivial null-space, and satisfies

$$G_z-G_w=(z-w)G_zG_w,$$ $$G_z^*=G_{\overline z},$$

for $z\in\mathbb C\setminus\mathbb R$. Consequently, $G_z$ is the resolvent of a self-adjoint operator $K^{\mathrm{s.a.}}$, regarded as the distinguished self-adjoint realization of $K$. The conjecture is intended to construct the resolvent and thereby the distinguished realization; the source presents it as an expectation requiring further assumptions.