Extension of the trace-representation formula for quantum Hellinger divergences

Let $\rho$ and $\sigma$ be the states appearing in the definition of the quantum Hellinger divergence $H_\alpha(\rho\|\sigma)$, and let $\alpha>1$. Trace-representation conjecture. For any $\alpha>1$,

$$H_\alpha(\rho\|\sigma)=\int_0^\infty \operatorname{Tr}\left(\rho(\sigma+s\mathds{1})^{-1}\right)^\alpha\,\mathrm{d}s-\frac{1}{\alpha-1}.$$

The paper proves this formula for integer $\alpha\geq2$; the conjecture extends that result to all real $\alpha>1$.