Plurality is Stablest conjecture for $k$ candidates

For $k\geq 2$, let $f\colon\{1,\ldots,k\}^{n}\to\Delta_{k}$, where $\Delta_k$ is the probability simplex, and let $\operatorname{Inf}_{i}(f_j)$ denote the influence of coordinate $i$ on the $j$th coordinate function. Define the noise stability by $S_\rho f=\sum_{j=1}^{k}S_\rho f_j$, and let $\mathrm{PLUR}_{k,n}$ be the plurality function, with ties mapped to the uniform vector $\frac1k\sum_{j=1}^{k}e_j$. For every $\rho\in[-1/(m-1),1]$ and $\varepsilon>0$, there is $\tau>0$ such that the following holds for every $f$ satisfying $\operatorname{Inf}_{i}(f_j)\leq\tau$ for all $i$ and $j$.

Plurality is Stablest conjecture. If $\rho\geq0$ and $\mathbb{E}f=\frac1k\sum_{j=1}^{k}e_j$, then

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

If $-1/(m-1)\leq\rho<0$, then

$$S_\rho f\geq\lim_{n\to\infty}S_\rho\mathrm{PLUR}_{k,n}-\varepsilon.$$

This is the formal discrete Plurality is Stablest statement cited as the source of the three-candidate conjecture and as the conjectural ingredient for sharp MAX-$k$-CUT hardness. The surrounding text notes that the three-candidate case is proved only for the narrower range $-.0234\leq\rho\leq.1082$, while the conjecture is expected for the stated range; the notation retains the source's parameter $m$.