Borda count is Stablest Conjecture

Let $S_k$ be the set of rankings of $k$ candidates, and let $f\colon S_k^n\to\Delta_k$ be a ranked-choice voting method, where $\Delta_k$ is the simplex of randomized outcomes on the candidates. For each coordinate function $f_j$, voter $i$ has influence $\operatorname{Inf}_i(f_j)$; the method is balanced when $\mathbb{E}f=\frac{1}{k}\sum_{i=1}^k e_i$, and it satisfies the Condorcet Loser Criterion when a candidate who loses pairwise to every other candidate cannot be selected. Let $S_\rho$ denote noise stability under independent ranking corruption, and let $\operatorname{Bor}_{k,n}$ be the Borda count method. Borda count is Stablest Conjecture. For any $k\geq2$, $\rho\in[0,1]$, and $\varepsilon>0$, there exists $\tau>0$ such that, for all $n\geq1$, if $\operatorname{Inf}_i(f_j)\leq\tau$ for every voter $i$ and candidate $j$, if $f$ is balanced, and if $f$ satisfies the Condorcet Loser Criterion, then

$$S_\rho f\leq\lim_{n\to\infty}S_\rho\operatorname{Bor}_{k,n}+\varepsilon.$$

This is the paper's formal ranked-choice analogue of Plurality is Stablest. The source does not report a resolution, so it remains open.