Asymptotic unfairness under convergent weight distributions

Let $m^{(n)}$ be a sequence of weight distributions with corresponding probability distributions $(P^{(n)})_{n \in \mathbb{N}}$, where $P^{(n)}=(p_i^{(n)})_{i \in \mathbb{N}}$ is a probability distribution on $\mathbb{N}$. Let $m^{(\infty)}$ be a weight distribution whose corresponding probability distribution $P^{(\infty)}=(p_i^{(\infty)})_{i \in \mathbb{N}}$ satisfies

$$\sup_{i \in \mathbb{N}}\left|p_i^{(n)}-p_i^{(\infty)}\right|\xrightarrow[n\to\infty]{}0.$$

Consider a sequence of $r$-splittings $m_{i_1^{(r)}}^{(n)},\ldots,m_{i_r^{(r)}}^{(n)}>0$ of a node $i$, with $m_{i_j^{(r)}}^{(n)}\to m_{i_j^{(r)}}^{(\infty)}$ for $j\in\\{1,2,\ldots,r\\}$, where the limiting terms form an $r$-splitting $m^{(\infty)}$. For a voting scheme with voting-power function $V_k^{(P)}(p)$ and the corresponding distributions and node weights after splitting, write $\widehat P_{r,i}^{(n)}$ and $\widehat p_{i_j^{(r)}}^{(n)}$ for the resulting distribution and probability, and similarly at infinity. Asymptotic unfairness conjecture. For every $k\in\mathbb{N}$,

$$\begin{aligned} \lim_{n\to\infty}\Bigg[\left(\sum_{j=1}^r V_k^{(\widehat P_{r,i}^{(n)})}(\widehat p_{i_j^{(r)}}^{(n)})\right)-V_k^{(P^{(n)})}(p_i^{(n)})\Bigg] &=\left(\sum_{j=1}^r V_k^{(\widehat P_{r,i}^{(\infty)})}(\widehat p_{i_j^{(r)}}^{(\infty)})\right)-V_k^{(P^{(\infty)})}(p_i^{(\infty)})\\>0. \end{aligned}$$

The claim asserts that the gain in voting power caused by the splitting remains strictly positive in the limit, so the voting scheme is not asymptotically fair for these convergent sequences of weight distributions. The supplied context discusses this as an asymptotic-unfairness phenomenon, but does not establish the displayed assertion or provide a resolution status.