The Gaussian-isoperimetric stability conjecture for hyperplanes

Equip $\mathbb{R}^n$ with the Gaussian metric. A hyperplane is a codimension-one affine subspace, and it is regarded as a solution of the isoperimetric problem when it separates the ambient space into regions of prescribed volume. Gaussian-isoperimetric stability conjecture. Every hyperplane in $\mathbb{R}^n$ endowed with the Gaussian metric is stable as a solution to the isoperimetric problem. Hyperplanes through the origin are known to have stability index $1$; the proposed extension concerns hyperplanes not passing through the origin.