Index-equating conjecture for UCB algorithms

Consider a $K$-armed bandit model. For arm $k$, let $N_{k,T}$ be the number of pulls by time $T$, let $\bar\mu_{k,T}$ be its empirical mean reward, and let $I(\cdot)$ be the index function of a UCB algorithm. In a “reasonable” bandit model and under a “reasonable” UCB algorithm, index-equating conjecture. The pull counts should satisfy

$$I\left(\bar\mu_{1,T},N_{1,T},T\right)\approx I\left(\bar\mu_{k,T},N_{k,T},T\right),\qquad 2\leq k\leq K,$$

with

$$\sum_{k=1}^{K}N_{k,T}=T.$$

This conjecture informally extends the equal-index characterization of the associated fluid system to the stochastic bandit process. The paper presents it as a natural conjecture within a perturbation-analysis framework; no resolution is supplied in the provided text.