Average-criterion PCL-indexability for restless bandits with imperfect feedback

Let $f(x,z)$ and $g(x,z)$ denote the average-criterion limiting marginal metrics, and define the diagonal MP index by

$$m(x)=\frac{f(x,x)}{g(x,x)}.$$

The average-criterion analogues of $\textup{(PCLI1)}$ and $\textup{(PCLI2)}$ are the corresponding positivity and monotonicity conditions for these metrics. Average-criterion PCL-indexability conjecture. The metrics $f(x,z)$, $g(x,z)$, and $m(x)$ satisfy the average-criterion analogues of $\textup{(PCLI1)}$ and $\textup{(PCLI2)}$ on the full threshold range, including the regime $x^1<z<x^0$.

This conjecture proposes that the structural properties established or suggested in tractable regimes persist for the average-cost problem, including the intermediate regime. The limiting formulas and regenerative representation provide computational evidence, but the full average-criterion verification remains open.