Kato's conjecture for positive Howland-Kato commutators

Let $P$ and $Q$ be the momentum and position operators, and let $f$ and $g$ be bounded, real, measurable functions. Define

$$K=i[f(P),g(Q)].$$

For a positive parameter $\alpha$, let $K_{\alpha}$ denote the class of functions having the representations given in the paper's equation (tanhrep), and let $\hat{\alpha}$ be the corresponding parameter for the second function. Kato conjecture. If $K\geq 0$ and $K\ne 0$, then $\pm f$ and $\pm g$ have the representations given in (tanhrep), with

$$\alpha\hat{\alpha}=\pi/2.$$

Equivalently, $\pm f\in K_{\alpha}$ and $\pm g\in K_{\hat{\alpha}}$. The paper proves substantial consequences of the assumptions, including absolute continuity and integrable derivatives, but the stated representation and analyticity conclusion is the conjectural part under discussion.