Plurality is stablest conjecture, discrete version

Let $m\geq 2$, let $\rho\in[0,1]$, and let $\varepsilon>0$. For a function $f:\{1,\ldots,m\}^n\to\Delta_m$, write $\operatorname{Inf}_i(f_j)$ for the influence of coordinate $i$ on component $f_j$, and let $S_\rho f$ denote its total noise stability. Let $e_1,\ldots,e_m$ be the standard basis vectors and let $\operatorname{PLUR}_{m,n}$ be the plurality function, with ties mapped to $m^{-1}\sum_{i=1}^m e_i$. Plurality is stablest conjecture, discrete version. There exists $\tau>0$ such that, whenever $\operatorname{Inf}_i(f_j)\leq\tau$ for every $i,j$ and $\mathbb{E}f=m^{-1}\sum_{i=1}^m e_i$,

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

This is the formal low-influence, equal-outcome-probability version of the voting claim and would identify plurality as asymptotically optimal for noise stability; it remains open in the full stated range.