Barthe's symmetric Gaussian surface-area minimizer conjecture

Fix $0<c<1$. For subsets $\Omega\subseteq\mathbb{R}^{n+1}$ satisfying

$$\Omega=-\Omega,\qquad \gamma_{n+1}(\Omega)=c,$$

consider the Gaussian surface area

$$\int_{\partial\Omega}\gamma_n(x)\,dx.$$

Barthe's conjecture. If $\Omega$ minimizes this quantity, then, after a rotation, there exist $r>0$ and $0\leq k\leq n$ such that

$$\partial\Omega=rS^k\times\mathbb{R}^{n-k}.$$

This asserts that solid round cylinders, including the relevant limiting cases and their complements, minimize Gaussian surface area among symmetric sets of prescribed Gaussian measure. The paper presents this as an open conjecture.