The Majority is Least Stable conjecture for linear threshold functions

Let $f\colon\{-1,1\}^{n}\rightarrow\{-1,1\}$ be a linear threshold function, with $n$ odd. For $\rho\in[0,1]$, let $\operatorname{Stab}_{\rho}[f]$ denote the noise stability of $f$ at correlation $\rho$, and let $\operatorname{Maj}_{n}$ be the majority function on $n$ variables. Majority is Least Stable conjecture. For all $\rho\in[0,1]$,

$$\operatorname{Stab}_{\rho}[f]\geq\operatorname{Stab}_{\rho}[\operatorname{Maj}_{n}].$$

This conjecture was proposed by Benjamini, Kalai, and Schramm and presented by Filmus et al. as an open problem. The paper's title and accompanying remark indicate that the conjecture as stated is false; independent counterexamples were observed by Sivakanth Gopi, and by Steven Heilman and Daniel Kane.