The boundary information inequality conjecture

Let $M$ be a manifold with boundary, let $f$ be a signal function, and let $K_M$ and $K_{\partial M}$ denote the physical information of $M$ and its boundary. Let $\tau$ be the restriction of the metric-like object $\sigma$ to $\partial M$, and define

$$K_{\partial M}=\int_{\partial M}\int_A\left\{\frac{\lVert\partial_\tau f\rVert_\tau^2}{f}-\frac{\lVert\psi\rVert_\tau^2}{f}\right\}.$$

Boundary information inequality conjecture.

$$K_M-K_{\partial M}\geq 0,$$

with equality for a critical choice of $f$.

The conjecture is motivated by analogous physical considerations for systems on finite domains or domains with boundary. No resolution is given in the source.