The inner-loop error-bound conjecture for unique information computation

Let $q$ and $\tilde q$ be the distributions produced by the inner-loop optimization, and let $\eta$ and $\tilde\eta$ be the corresponding expectation parameters for the marginal distributions on $Y$ and $Z$. The quantities $\|\tilde q-q\|_\infty$ and $\|\tilde\eta-\eta\|_1$ measure the approximation errors in the distribution and expectation parameters, respectively.

Inner-loop error-bound conjecture.

$$\|\tilde q-q\|_\infty \leq \|\tilde\eta-\eta\|_1.$$

This bound would provide the criterion needed to interrupt the inner iteration while guaranteeing a prescribed accuracy for the computed distribution. The supplied text does not indicate whether the claim has been proved or remains open.