Global monotonicity conjecture for the Kubo–Mori update

Let $X$ be the observed evidence, let $\mathcal{E}$ be the quantum channel, and let $\sigma$ be the current density matrix. Write $\mathcal{L}_X(\sigma)$ for the log-likelihood and define the full Kubo–Mori update by

$$\sigma_+=\mathcal{R}^{\text{KM}}_{\sigma,\mathcal{E}}(X).$$

Global monotonicity conjecture. The Kubo–Mori update can only increase the log-likelihood:

$$\mathcal{L}_X(\sigma_+)\geqslant\mathcal{L}_X(\sigma).$$

Equality holds if and only if $\sigma$ is a fixed point of the Kubo–Mori update, namely $\sigma_+=\sigma$.

The preceding local increase theorem establishes strict improvement for sufficiently small step sizes whenever the Riemannian gradient is nonzero. The conjecture asserts the stronger global statement for the full multiplicative update with step size $\eta=1$; its resolution is not supplied in the source.