Plurality Is Asymptotically Stablest conjecture

Let $0\leq\rho\leq1$ and let $q\geq2$ be an integer. For $f:[q]^n\to[0,1]$, let $\mathbb{S}_\rho(f)$ denote its noise stability and let $\mathrm{Inf}_i(f)$ denote the influence of coordinate $i$. Plurality Is Asymptotically Stablest conjecture. For every $\epsilon>0$ there exists $\tau>0$ such that, if ${\bf E}[f]=1/q$ and $\mathrm{Inf}_i(f)\leq\tau$ for all $i$, then

$$\mathbb{S}_\rho(f)\leq q^{-2/(1+\rho)+o(1)}+\epsilon,$$

where $o(1)$ tends to $0$ as $q\to\infty$, independently of $\epsilon$ and $\tau$. This is an asymptotic stability conjecture for low-influence functions on the $q$-ary cube, motivated by optimal approximation results. The source does not state a resolution.