Time monotonicity conjecture for Bayesian sequential composite hypothesis testing

Let $V(n,\pi)$ be the value function defined in, where $n$ denotes the time index, $\pi$ is the relevant state variable, $\mu$ is a prior distribution, and $\nu$ is the parameter appearing in the testing problem. Time monotonicity conjecture. The function $V(n,\pi)$ is non-decreasing in $n$ for any prior distribution $\mu$ and any $\nu$. The preceding results establish time monotonicity only under restrictive conditions, while the authors report no counterexamples within the exponential family and verify the property in several particular examples; the conjecture asserts it for arbitrary prior distributions.