Conjecture on nonconvergence and asymptotic uniformity for vanishing-duration partitions

Let $T_\infty=(h_1,h_2,\ldots)$ be a fixed partition with $h_1+h_2+\cdots=+\infty$, and let $v_{T_\infty,\lambda}(p)$ denote the value of the game from Theorem~ with discount parameter $\lambda$ and initial probability vector $p$. The notation $\sup h_i$ denotes the supremum of the stage durations. Nonconvergence and asymptotic uniformity conjecture. For every fixed partition $T_\infty$, the pointwise limit

$$\lim_{\lambda\to 0}v_{T_\infty,\lambda}(p)$$

does not exist. Moreover,

$$\left|\limsup_{\lambda\to 0}v_{T_\infty,\lambda}(p)-\liminf_{\lambda\to 0}v_{T_\infty,\lambda}(p)\right|\to 0$$

as $\sup h_i$ tends to $0$, uniformly in $p$. This conjecture strengthens the preceding results: for the game considered, the uniform vanishing-stage-duration limit exists whereas the pointwise limit for a particular partition does not. It predicts that every fixed partition retains pointwise nonconvergence, but that the oscillation between the limiting upper and lower values disappears uniformly as all stage durations become small.